text:
\section{Introduction}
The derived category of a non-semisimple symmetric algebra is a beast.  For example, a simple classification of perfect objects usually does not exist, which makes the task of finding derived equivalent algebras - one of the central themes in modular representation theory of finite groups, extremely difficult.  The Okuyama-Rickard construction \cite{Oku,Ric} gives an easy way to calculate non-trivial tilting complexes.  Therefore, one would naturally hope that all tilting complexes can be obtained by applying such a  construction repeatedly, and  use this to determine, say, the derived equivalence class.  Roughly speaking, an algebra is tilting-connected if such a hope holds.

One of the main results in \cite{AI} is that there is a partial order structure on the set of tilting complexes.  Using this translation, tilting-connectedness simply means that the Hasse quiver of this partially ordered set is connected.  One could attempt to exploit this combinatorial property to better understand, for instance, the derived Picard group(oid) of the derived category.  See \cite{Zvo2} for a fruitful result in this direction.

Determining tilting-connectedness is not easy - at least with the current technology.  So far, the only known tilting-connected symmetric algebras are the local ones \cite{AI}, and the representation-finite ones \cite{A}.  The ``proof" for tilting-connectedness of local symmetric algebras in fact is the answer we wanted originally - a classification of tilting complexes.  More precisely, a tilting complex of a local algebra is precisely a stalk complex given by a finite direct sum of the (unique) indecomposable projective module.  On the other hand, the proof for the representation-finite case does not generalise to an arbitrary finite dimensional symmetric algebra.

A symmetric algebra is said to be tilting-discrete if the set of $n$-term tilting complexes is finite for any natural number $n$.  In such a case, the algebra will be tilting-connected.  This notion is introduced in \cite{A} in order to find a more computable approach to determine tilting-connectedness.  To see if this approach can be successful for non-representation-finite non-local symmetric algebras, we use the Brauer graph algebras as a testing ground.  We remark here that the tilting-discreteness, along with its applications, of the preprojective algebras is also studied in a parallel work of the second author and Mizuno \cite{AM}.

The representation-finite Brauer graph algebras (the Brauer tree algebras) was discovered by Brauer in the forties during the dawn of modular representations of finite groups.  The generalisation - Brauer graph algebras - is the main subject of interest in this paper.
The Brauer graph algebras are tame symmetric algebras which are arguably the easiest class of symmetric algebras that one can play with.  This is because the structure of such an algebra is encoded entirely in a simple combinatorial object, called the Brauer graph.  It often turns out that one can replace many algebraic and homological calculations into simple combinatorial games on the Brauer graph.  These results in turn would give inspirations to develop techniques and theories for larger classes of algebras, such as group algebras or tame symmetric algebras.  We hope to apply this philosophy to flourish the notions of tilting-connectedness and tilting-discreteness.

The first homological calculation which will be turned into pure combinatorics is the determination of two-term (pre)tilting complexes (Theorem \ref{mainbij}).  Since the combinatorics is entirely new, we avoid giving the statement of the theorem here.
For readers with no knowledge about Brauer graphs, we briefly remind that a Brauer graph is a graph with an ordering of edges around each vertex, and a positive number called multiplicity associated to each vertex.  Forgetting the multiplicities, one obtains an orientable ribbon graph (or fatgraph) - a combinatorial object which appears in many other areas of mathematics, such as dessin d'enfants, Teichm\"{u}ller and moduli space of curves, homological mirror symmetry, etc.

Our second result is the classification of tilting-discrete Brauer graph algebras.  We achieve this by using the ``two-term combinatorics" and the tilting mutation theory for Brauer graph algebras.  Since most of the tilting-discrete Brauer graph algebras are neither representation-finite nor local, we have a new class of tilting-connected symmetric algebras.

\begin{theorem}
Let $\mathbb{G}$ be a Brauer graph, and $\Lambda_\mathbb{G}$ its associated Brauer graph algebra.
\begin{enumerate}[(1)]
\item (Proposition \ref{2tilt}) The partially ordered set of basic two-term tilting complexes of $\Lambda_\mathbb{G}$ is isomorphic to that of $\Lambda_{\mathbb{G}'}$ if $\mathbb{G}$ and $\mathbb{G}'$ have the same underlying ribbon graph.

\item (Theorem \ref{TD of type odd}) $\Lambda_\mathbb{G}$ is tilting-discrete if, and only if, $\mathbb{G}$ contains at most one cycle of odd length, and no cycle of even length.

\item (Corollary \ref{cor-type-odd-equiv}) In the case of (2), any algebra derived equivalent to  $\Lambda_\mathbb{G}$ is also a Brauer graph algebra $\Lambda_{\mathbb{G}'}$.  Moreover, $\mathbb{G}'$ is flip equivalent to $\mathbb{G}$ in the sense of \cite{A2}.
\end{enumerate}
\end{theorem}
Note that the derived equivalence class in (3) is not entirely new; for slightly more details, see the discussions at the end of Section \ref{sec-tilt-discrete}.

The classification and description of two-term tilting complexes (and their indecomposable summands) of the Brauer star algebras, i.e. (representation-finite) symmetric Nakayama algebras, have already been studied in \cite{SZI}, and implicitly in \cite{Ada}.  In the case of the Brauer tree algebras, these tasks were carried out in \cite{AZ1,Zvo1}.  We note that the result in \cite{SZI} actually classifies more than just two-term tilting complexes.  In all the mentioned articles, as well as ours, the indecomposable summands of two-term tilting complexes are classified first. Then one gives the conditions on how they can be sum together to form tilting complexes.  In contrast to \cite{SZI,Zvo1}, we will not calculate the corresponding endomorphism algebra in this article, but leave it for our sequel.  For Brauer tree algebras, the classification in \cite{Zvo1} is described in terms of the Ext-quiver of the algebra, whereas ours are described by combinatorics on the defining ribbon graph.

The combinatorial language we use is heavily influenced by the ribbon graph theory.  The use of this language for Brauer graph algebras is slightly different from the traditional approach used in, for example, \cite{Ro,Kau}, but it is also not new - we learn it from the paper of Marsh and Schroll \cite{MS}, which relates Brauer graphs with surface triangulation (and $n$-angulation) and cluster theory.  The key advantage of adopting this approach is that we can clear out many ambiguities when there is a loop in the Brauer graph.  Moreover, while writing up this article, this new language gives us a glimpse of a connection between geometric intersection theory and the tilting theory of Brauer graph algebras (see Remark \ref{rmk-geometry}).  We hope to address this issue in a subsequent paper.

Our approach to the homological calculations draws results and inspirations from \cite{AI,A,AIR}.  The key technique which allows the classification of two-term (pre)tilting complexes possible is Lemma \ref{lem-non-zero-hom}.  This is a particular case of a vital lemma in relating $\tau$-tilting theory with two-term silting/tilting complexes in \cite{AIR}.  This homological property was also used in \cite{AZ1,Zvo1}.  While we will not mention and introduce $\tau$-tilting theory formally, a partial motivation of this work is in fact to obtain a large database of calculations for $\tau$-tilting theory, and hopefully to inspire further development in the said theory.  The idea of investigating tilting-discreteness using only knowledge about two-term tilting complexes in this article (and the investigation of similar vein in \cite{AM}) is also inspired by the elegance of $\tau$-tilting theory.

This article is structured as follows.  We will first go through in Section \ref{sec-ribbon-combinatorics} the essential ribbon (and Brauer) graph combinatorics needed for this paper.  There is no mathematical prerequisite in order to understand this section, although knowledge of basic notions in ordinary graph theory would be helpful to find intuitions.
Section \ref{sec-alg-hom-prelim} is devoted to recall known results and notions needed to understand the algebraic and homological side of the first main theorem.
A brief reminder on the basic tilting theory is separated out in Subsection  \ref{sec-tilting-theory}, whereas the review on Brauer graph algebras and their modules can be found in Subsection \ref{sec-BGA}.
In Section \ref{sec-twotilt}, we will first explain some elementary observations on the two-term tilting complexes of a Brauer graph algebra.  This allows us to write down maps between the homological objects (two-term tilting complexes and its direct summands) and the combinatorial objects, which form the statement of our first main result (Theorem \ref{mainbij}).

We will spend the entire Section \ref{sec-proof} to prove Theorem \ref{mainbij}.
In Section \ref{sec-application}, we explore an application of Theorem \ref{mainbij}.
We first present some preliminary material on tilting-connectedness and related notions in Subsection \ref{sec-application-prelim}.
Then we determine a sufficient condition for a Brauer graph algebra to be tilting-connected in Subsection \ref{sec-tilt-discrete}.

Finally, we provide additional material which are not directly related to the main result of this article.
In Appendix \ref{sec-appendix-I}, we compare the ribbon combinatorics used throughout this article and the string combinatorics used in other studies of Brauer graph algebras.
Appendix \ref{sec-app-endo} explains why the endomorphism ring of an irreducible mutation (see terminologies from Section \ref{sec-application-prelim}) of a Brauer graph algebra is symmetric special biserial.
This result is needed in the proof of \cite[Theorem 3.1]{A2} (Proposition \ref{prop-flip der equiv}) which is not shown properly due to a false-proof in \cite{Pog} (cf. \cite[Appendix]{AIP}).
For the ease of readers, we include the list of notations used in this article and the locations of their first appearances in Appendix \ref{sec-notations}.

\section*{Acknowledgment}
We owe our deepest gratitude to Osamu Iyama for many fruitful discussions, as well as the financial support for the third author's visit to Nagoya University, which nurtured this research.  This article is typed up during AC's subsequent visits at Nagoya University, and finished during the first author's visit at Uppsala University.  We are thankful for the hospitality of these institutions.
We thank Ryoichi Kase for informing us various minor mistakes in an earlier version of this script.
AC would also like to thank Alexandra Zvonareva for discussions which inspired some of the combinatorial results.

\section*{Convention}
The following assumptions and conventions will be imposed throughout the article.
\begin{enumerate}[(1)]
\item For a sequence $w=(e_1,e_2,\ldots,e_n)$, a subsequence $w'$ is said to be \emph{continuous}, and denoted by $w'\subset w$\label{sym-subseq}, if it is of the form $(e_i,e_{i+1},\ldots, e_{i+k})$.
We adopt this convention to avoid any possible confusion with string combinatorics (cf. Appendix \ref{sec-appendix-I}).
\item The composition of maps $f:X\to Y$ and $g:Y\to Z$ is $gf:X\to Z$.
\item All algebras are assumed to be basic, indecomposable, and finite dimensional over an algebraically closed field $\Bbbk$.  We will often use $\Lambda$ to denote such an algebra.
\item We always work with finitely generated right modules, and use $\mod\Lambda$ to denote the category of finitely generated right $\Lambda$-modules.
\item We denote by $\operatorname{proj}\Lambda$ the full subcategory of $\mod\Lambda$ consisting of
all finitely generated projective $\Lambda$-modules.
The bounded homotopy category of $\operatorname{proj}\Lambda$ is denoted by $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$.
\item We sometimes write $\Lambda=\Bbbk Q/I$, where $Q$ is a finite quiver with relations $I$.
\item A simple (resp. indecomposable projective) $\Lambda$-module corresponding to a vertex $i$ of $Q$ is denoted by $S_i$\label{sym-simple} (resp. by $P_i$\label{sym-proj}); an arrow of $Q$ is identified with a map between indecomposable projective $\Lambda$-modules.
\item For an object $X$ (in $\mod \Lambda$ or $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$), we denote by $|X|$\label{sym-cardinal} the number of isomorphism classes of the indecomposable direct summands of $X$.
\end{enumerate}

\section{Ribbon graphs combinatorics}\label{sec-ribbon-combinatorics}
\begin{definition}
A \emph{ribbon graph}\label{sym-G} is a datum $\mathbb{G}=(V,H,s,\overline{\,\cdot\,},\sigma)$, where
\begin{enumerate}
\item $V$\label{sym-V} is a finite set, where elements are called \emph{vertices}.
\item $H$\label{sym-H} is a finite set, where elements are called \emph{half-edges}.
\item $s:H\to V$\label{sym-s} specifies the vertex $s(e)$ for which a half-edge $e$ is  emanating.
\item $\overline{\,\cdot\,}:H\to H$\label{sym-ol} is a fixed-point free involution, i.e. $\overline{e}\neq e$ and $\overline{(\,\overline{e}\,)}=e$ for all $e\in H$.
\item $\sigma:H\to H$\label{sym-sigma} is a permutation on $H$ so that the set of $\langle\sigma\rangle$-orbits is $V$ and $s:H\to V=H/\langle\sigma\rangle$ is the induced projection.
For $v\in V$, the \emph{cyclic ordering around $v$} is the restriction of $\sigma$ on $s^{-1}(v)$, which will be denoted by $(e,\sigma(e),\ldots,\sigma^{k-1}(e))_v$\label{sym-cyclic-order-v} for some $e\in H$ with $s(e)=v$, where $k=|s^{-1}(v)|$.
Viewing this notation as a sequence up to cyclic permutation, its subsequences will be called \emph{cyclic suborderings around $v$}.
\end{enumerate}
\end{definition}

The geometric realisation of $\mathbb{G}$ is $(H\times[0,1])/\sim$, where $\sim$ is the equivalence relation defined by
$(e,\ell)\sim(\overline{e},1-\ell)$ and $(e,0)\sim (\sigma(e),0)$.
Intuitively, we ``glue" $e$ and $\overline{e}$ together to form a graph-theoretic edge (geometric line segment) incident to
$s(e)$ and $s(\overline{e})$.
Because of this, we call an unordered pair $\{e,\overline{e}\}$ an \emph{edge} of $\mathbb{G}$, and the vertices (or vertex) $s(e)$ and $s(\overline{e})$ the \emph{endpoint(s)} of the edge $E$.
We also define the \emph{valency} of a vertex $v$ as $\mathrm{val}}\newcommand{\vir}{\mathrm{vr}(v):=|s^{-1}(v)|$\label{sym-val}.
Note that a ribbon graph (or its geometric realisation) is also called a locally embedded graph or a fatgraph in the literature.

A \emph{Brauer graph} is a ribbon graph equipped with a \emph{multiplicity function} $\operatorname{\mathfrak{m}}:V\to \mathbb{Z}_{>0}$\label{sym-mm}
which assigns a positive integer, called \emph{multiplicity}, to each vertex of $\mathbb{G}$.

We will impose the following assumptions and conventions on ribbon graphs throughout.
\begin{itemize}
\item A Brauer/ribbon graph is always assumed to be connected, i.e. its geometric realisation is connected.

\item We will always notate half-edges by small alphabets such as $e\in H$,
and we will notate the corresponding edge $\{e,\overline{e}\}$ by the same letter in capital, which is $E$ here.

\item Due to the definition of Brauer graph algebras (see Definition \ref{def-BGA}),
we usually present $\mathbb{G}$ graphically using its geometric realisation with the cyclic ordering of (half-)edges around
each vertex presented in the counter-clockwise direction.

\item Having said that, whenever we present a local structure such as
\begin{align}
\xymatrix{
&  {v} \ar@{-}[r]^{e_2}\ar@{-}[l]_{e_1}&
}\notag
\end{align}
then the two lines ($e_1$ and $e_2$) emanating from $v$ are regarded as half-edges emanating from $v$.
\end{itemize}

\begin{example}\label{eg-two-loop}
Let $\mathbb{G}$ be the ribbon graph:
\[
(\{v\},\;\; \{e,\overline{e},f,\overline{f}\},\;\; s\equiv v,\;\; \overline{\,\cdot\,},\;\;(e,\overline{e},\overline{f},f)_v),
\]
Then the geometric realisation of $\mathbb{G}$ is:
\begin{align}
\xymatrix@R=30pt@C=30pt{ & & & \\
&  {v} \ar@{-} `ul[l] `dl[d]_{E}  `dr[] []  \ar@{-} `ur[r] `dr[d]^{F}  `dl[] []
& &\text{or locally around $v$: }  \\
& & &} \xymatrix@R=25pt@C=40pt{ & &\\
 &  {v} \ar@{-}[lu]^{e} \ar@{-}[ld]^{\overline{e}}  \ar@{-}[rd]^{\overline{f}} \ar@{-}[ru]^{f} & \\
& & } \notag
\end{align}
\end{example}

\begin{definition}[Half-walk, walk, and signed walk]\label{def-walks}
\begin{itemize}
\item[(a)] A non-empty sequence $w$\label{sym-halfwalk} of half-edges $(e_1,\ldots,e_l)$ such that
$s(e_{i+1})=s(\overline{e_i})$ for all $i=1,\ldots,l-1$ is called a \emph{half-walk}.
Defining $\overline{w}=(\overline{e_l},\ldots,\overline{e_1})$ makes $\overline{\,\cdot\,}$ an involution on the set of half-walks.

\item[(b)] A \emph{walk} on a ribbon graph $\mathbb{G}$ is the unordered pair $W=\{w,\overline{w}\}$ of half-walks.  For exposition convenience, by ``a walk $W$ given by (the half-walk) $w$" we mean that $W=\{w,\overline{w}\}$\label{sym-walk}.

\item[(c)] For a half-walk $w=(e_1,\ldots,e_l)$, we define $s(w):=s(e_1)$ and $s(\overline{w})=s(\overline{e_l})$
as the \emph{endpoint(s)} of $W=\{w,\overline{w}\}$.  We say that a vertex $v$ (resp. half-edge $e$) is in $W$ if $v=s(\overline{e_l})$ or $v=s(e_i)$ (resp. $e=e_i$ or $\overline{e_{i}}$) for some $i\in\{1,\ldots,l\}$.

\item[(d)] A \emph{signature} on a walk $W=\{w=(e_1,\ldots,e_l),\overline{w}\}$ is an assignment of signs $\epsilon_W(e)=\epsilon_W(\overline{e})$\label{sym-signature} on the half-edges in $W$ such that $\epsilon_W(e_i)\neq \epsilon_{W}(e_{i+1})$ for all $i\in\{1,\ldots,l-1\}$.
A walk equipped with a signature is called a \emph{signed walk}. A \emph{signed half-walk} is a half-walk $w$ equipped with a signature on $W=\{w,\overline{w}\}$.  We often denote a signed half-walk by $(w;\epsilon)$\label{sym-signed-hw} or $(e_1^{\epsilon(e_1)},e_2^{\epsilon(e_2)},\ldots,e_l^{\epsilon(e_l)})$.
Denote by $\operatorname{\mathsf{SW}}(\mathbb{G})$\label{sym-SWG} the set of signed walks of $\mathbb{G}$.
\end{itemize}
\end{definition}
Note that the notation of half-walk has no subscript next to the closing bracket, whereas the subscript $v$ in $(e_1,e_2,\ldots,e_k)_v$ clarifies that this sequence represents a cyclic subordering around $v$.

The combinatorial gadget is named in the spirit of the Green's walk around a Brauer tree \cite{Gre}, which is essentially a combinatorial description of the minimal projective resolutions of what we call the hook modules (Definition \ref{def-hook}) of a Brauer tree algebra.  We will see in Section \ref{sec-twotilt} that the signed walks coincide with the minimal projective \emph{presentations} of certain modules of a Brauer graph algebra.

Another reason why we use the name ``walk" in the above definition is that they can be visualised as walks in graph theory,
when we regard (the geometric realisation of) $\mathbb{G}$ as a graph of undirected edges.
However, they are not exactly the same as graph-theoretic walks, as we can see in the following example.

\begin{example}
Let $\mathbb{G}$ be the ribbon graph in the previous example.
There are four signed walks induced by a graph-theoretic walk (with signs) $(v,E^+,v,F^-,v)$.
They are given by half-walks $w_1=(e^+,f^-)$, $w_2=(e^+,\overline{f}^-)$, $w_3=(\overline{e}^+,f^-)$, $w_4=(\overline{e}^+,\overline{f}^-)$.

Note that one can not always find a signature for a walk.  For example, the walk $(e,\overline{e})$ in this example cannot have a signature.  On the other hand, if one can define a signature on a walk, then there are exactly two choices of signature.
\end{example}

\begin{definition}[Common subwalk] \label{def-common-subwalk}
\begin{enumerate}
\item We denote by $w\cap w'$\label{sym-common-subwalk} the set of half-walks $z$ such that $z\subset w$, $z\subset w'$, and
there is no $z'\neq z$ with $z\subset z'\subset w$ and $z\subset z'\subset w'$.

\item A \emph{subwalk} $Z$ of $W$ is a walk given by some continuous subsequence $z\subset w\in W$.

\item A \emph{common subwalk} $Z$ of $W$ and $W'$ is a walk given by $z$ with $z\subset w$ and $z\subset w'$ for some $w\in W$ and $w'\in W$.

\item We denote by $W\cap W'$\label{sym-set-common-subwalk} the set of common subwalks $Z$ given by $z\in w\cap w'$, for some $w\in W$ and $w'\in W'$.  An element of $W\cap W'$ is called a \emph{maximal common subwalk} of $W$ and $W'$.
\end{enumerate}
\end{definition}

To ease our burden of explaining the combinatorics for various definitions and proofs,
we will attach some extra data to a signed half-walk $w=(e_1,\ldots,e_l;\epsilon_W)\in W$
which are uniquely determined by the signature of $W$.

\begin{definition}
A \emph{virtual (half-)edge} is an element in the set $\{\vir_-(e),\vir_+(e)\mid e\in H\}$\label{sym-vir}.  We can augment $s$ on the virtual edges so that $s(\vir_\pm(e))=s(e)$.
Let $v$ be $s(e)$ and suppose $(e_1,\ldots,e_k)_v$ is the cyclic ordering around $v$.
We define the \emph{cyclic ordering (around $v$) accounting the virtual edges} as
\begin{align}\label{eq-cyc-ord-vir}
(\vir_-(e_1),e_1,\vir_+(e_1),\vir_-(e_2),e_2,\vir_+(e_2),\ldots,\vir_-(e_k),e_k,\vir_+(e_k))_v.
\end{align}
A (possibly non-continuous) subsequence of (\ref{eq-cyc-ord-vir}) is called \emph{cyclic subordering around $v$ accounting the virtual edges}.
Suppose  a signed walk $W$ is given by $w=(e_1,\ldots,e_l; \epsilon_W)$.  We define the following virtual edges attached to $W$:
\begin{align}
e_0 = \overline{e_0} & := \vir_{-\epsilon_W(e_1)}(e_1) \notag \\
e_{l+1} = \overline{e_{l+1}} & := \vir_{-\epsilon_W(e_l)}(\overline{e_l})\notag
\end{align}
We also define $\epsilon_W(e_0)=\epsilon_W(\overline{e_0})=-\epsilon_W(e_1)$ and $\epsilon_W(e_{l+1})=\epsilon_W(\overline{e_{l+1}})=-\epsilon_W(e_l)$.
\end{definition}

From now on, one should always bear in mind the following convention:
\emph{
\begin{center}
A cyclic subordering around an endpoint of a (half-)walk involving $e_0$ and/or $e_{l+1}$ is always regarded as the cyclic subordering accounting the virtual edges.
\end{center}
}
Unless otherwise specified, we fix $W$ and $W'$ as two (not necessarily distinct) signed walks given by half-walks
$w=(e_1,\ldots,e_m;\epsilon_W)$ and $w'=(e_1',\ldots,e_n';\epsilon_{W'})$ respectively.
Moreover, it is automatically understood what we mean by $e_0,e_{m+1},e'_0,e'_{n+1}$, etc. from the definition of virtual edges.

\begin{definition}[Sign condition]
We say that a signed walk $W$ (or a signed half-walk $w\in W$) \emph{satisfies the sign condition} if $\epsilon_W(e_1)=\epsilon_W(e_m)$ whenever $s(e_{1})=s(\overline{e_{m}})$.
In general, we say that two walks $W, W'$ satisfy the sign condition if all of the following conditions are satisfied:
\begin{itemize}
\item $\epsilon_W(e_1)=\epsilon_{W'}(e'_1)$ if $s(e_1)=s(e'_1)$;
\item $\epsilon_W(e_1)=\epsilon_{W'}(e'_n)$ if $s(e_1)=s(\overline{e'_n})$;
\item $\epsilon_W(e_m)=\epsilon_{W'}(e'_1)$ if $s(\overline{e_m})=s(e'_1)$;
\item $\epsilon_W(e_m)=\epsilon_{W'}(e'_n)$ if $s(\overline{e_m})=s(\overline{e'_n})$.
\end{itemize}
\end{definition}

To improve readability of various definitions, statements, and proofs, we only write down the half-edges required in the cycle of the cyclic ordering around their emanating vertex.  In other words, if, for instance, the half-edges $e,f,$ and $g$ are the only important half-edges around $v$ in a definition (like the following one), then we will write $(e,f,g)_v$ instead of $(e,\ldots,f,\ldots,g,\ldots,\sigma^{-1}(e))_v$.

\begin{definition}[Non-crossing condition at a maximal common subwalk]
Let $Z$ be a maximal common subwalk of $W$ and $W'$ given by $z=(t_1,t_2,\ldots, t_\ell)\in w\cap w'$ so that $t_{k}=e_{i+k-1}=e'_{j+k-1}$ for all $k\in\{1,\ldots,\ell\}$.
Then we call the pair of cyclic suborderings on $\{\overline{e_{i-1}},\overline{e_{j-1}'},t_1\}$ and $\{e_{i+\ell},e_{j+\ell},\overline{t_\ell}\}$ the \emph{neighbourhood cyclic orderings} of $Z$.
We say that $W$ and $W'$ are \emph{non-crossing at $Z$} if the following holds:
\begin{itemize}
\item[(NC1)] $\epsilon_W(t_{k})=\epsilon_{W'}(t_{k})$ for each $k\in\{ 1,\ldots, \ell\}$.
\item[(NC2)] With the exception of $i=j=1$ and/or $m+1-i-\ell=n+1-j-\ell=0$ (i.e. $u$ and/or $v$ being the endpoint(s) of \emph{both} walks), the neighbourhood cyclic orderings of $Z$ are
\begin{align}
\textnormal{either } &(t_1,\overline{e_{i-1}},\overline{e'_{j-1}})_{s(z)}\ \textnormal{ and }\ (\overline{t_\ell},e'_{j+\ell},e_{i+\ell})_{s(\overline{z})}\ \textnormal{ respectively}, \notag\\
\textnormal{or }\ &(t_1,\overline{e'_{j-1}},\overline{e_{i-1}})_{s(z)}\ \textnormal{ and }\ (\overline{t_\ell},e_{i+\ell},e'_{j+\ell})_{s(\overline{z})} \ \textnormal{ respectively}. \notag
\end{align}

Visualising the two cases locally:
\begin{align}
\xymatrix@C=15pt{
 \ar@{-}[rd]_{\overline{e_{i-1}}} & & & & & & \ar@{-}[ld]^{e_{i+\ell}} \\
  &  {u} & \ar@{-}[l]_{t_{1}} \ar@{.}[rr]& &\ar@{-}[r]^{\overline{t_\ell}} &  {v}\\
\ar@{-}[ru]^{\overline{e'_{j-1}}} & & & & & &\ar@{-}[lu]_{e'_{j+\ell}}
}\qquad \xymatrix@C=15pt{
 \ar@{-}[rd]_{\overline{e'_{j-1}}} & & & & & & \ar@{-}[ld]^{e'_{j+\ell}} \\
  &  {u} & \ar@{-}[l]_{t_{1}} \ar@{.}[rr]& &\ar@{-}[r]^{\overline{t_\ell}} &  {v}\\
\ar@{-}[ru]^{\overline{e_{i-1}}} & & & & & &\ar@{-}[lu]_{e_{i+\ell}}
}\notag
\end{align}
\end{itemize}
\end{definition}
\medskip

For a vertex in $v=s(e_i)$ in $W$, we refer to the set $\{\overline{e_{i-1}},e_i\}$ as a \emph{neighbourhood} of $v$ in $W$.
Suppose $\{a,b\}$ and $\{c,d\}$ are neighbourhoods of $v$ in $W$ and in $W'$ respectively.
We say that $v$ is an \emph{intersecting vertex} of $W$ and $W'$, if $a,b,c,d$ are pairwise distinct.
Note that $v$ can be an intersecting vertex with respect to multiple different pairs of neighbourhoods.

\begin{definition}[Non-crossing condition at an intersecting vertex and admissible walks]\label{def-NC-admissible}
We say that $W$ and $W'$ is \emph{non-crossing at the intersecting vertex $v$} if the following condition is satisfied.
\begin{itemize}
\item[(NC3)] If $v$ is an intersecting vertex with respect to the neighbourhoods $\{a,b\}$ in $W$ and $\{c,d\}$ and $W'$, and at most one of $a,b,c,d$ is virtual, then we have either one of the following cyclic subordering around $v$ and signature:
\begin{align}
\notag (a^+,b^-,c^+,d^-)_v, \quad \;\; (a^+,b^-,c^-,d^+)_v.
\end{align}
The local structure around $v$ for these two conditions can be visualised as
\begin{align}
\xymatrix@R=20pt@C=30pt{
\ar@{-}[dr]_{b^-} & & \ar@{-}[dl]^{a^+} & & \ar@{-}[dr]_{b^-} & & \ar@{-}[dl]^{a^+}\\
 &  {v} &  & \text{and} &  &  {v} &  \\
\ar@{-}[ur]^{c^+} & & \ar@{-}[ul]_{d^-} & & \ar@{-}[ur]^{c^-} & & \ar@{-}[ul]_{d^+}
}\notag
\end{align}
respectively.
\end{itemize}

We say that two walks $W,W'$ are \emph{non-crossing}, or they satisfy the \emph{non-crossing conditions}, if they are non-crossing at all maximal common subwalks and all intersecting vertices.

If in addition $W=W'$, we may specify that $W$ is \emph{self-non-crossing}.  An \emph{admissible walk} is a self-non-crossing (signed) walk which also satisfies the sign condition.  Denote by $\operatorname{\mathsf{AW}}(\mathbb{G})$\label{sym-AWG} the subset of $\operatorname{\mathsf{SW}}(\mathbb{G})$ consisting of all admissible walks.
\end{definition}
\begin{remark}\label{rmk-geometry}
While it is easy to see why (NC2) is called non-crossing, it is not apparent that why (NC1) is a non-crossing condition but not a sign condition, and why such signatures on half-edges around $v$ are required in (NC3) to make them non-crossing.  Although we will not use this representation of signed walks in this article, the correct graphical (geometrical) realisation is as follows.  When we go along the signed (half-)walk, say $(e_i,e_{i+1})\subset w$ with $i\neq 0$ and $i\neq m+1$, instead of visualising the situation as a line passing through a vertex:
\begin{align}
\xymatrix@C=40pt{ &  {v} \ar@{-}[l]_{\overline{e_i}} \ar@{-}[r]^{e_{i+1}}&}
\notag
\end{align}
we think of the vertex $v=s(e_{i+1})$ as lying below (resp. above) the line (relative to this presentation) if $\epsilon_W(e_{i+1})=-$ (resp. $\epsilon_W(e_{i+1})=+$).

This ``correct" visualisation is in fact a generalisation of the technique used in \cite{KS,ST} - from a disc to compact orientable Riemann surfaces with marked point and boundaries (specified by $\mathbb{G}$).  In this geometric setting, our signed walks become certain type of curves, and the non-crossing condition translates into requiring a curve to have no self-intersection.  We hope to explain this geometric setting with more details in a sequel paper.
\end{remark}

Checking non-crossing condition is in practice much easier than the way it is defined here - draw the walk around the geometric realisation of $\mathbb{G}$ in a non-crossing way and put signs on the edges to check (NC3).  We briefly explain how to check non-crossing-ness algorithmically here, and recommend the reader to carefully go through the proof of the next proposition in order to familiarise the procedure.

First, fix some $w\in W$ and $w'\in W'$.  Start with a vertex, say $v=s(e_i)=s(e'_j)$, which is in both $w$ and $w'$.  In the special case of $W=W'$, one can simply start with a vertex which appears at least twice in (any) $w\in W$.  If the half-edges $\overline{e_{i-1}},e_i,\overline{e_{j-1}'},e_j'$ are pairwise distinct, then $v$ is an intersecting vertex with respect to the neighbourhoods given by these half-edges.
One can verify if the signed cyclic ordering required by (NC3) is satisfied simultaneously.
Otherwise, we have a common subwalk given by the coinciding half-edge, and one can expand this half-edge to some $z\in w\cap w'$ or $z\in w\cap \overline{w'}$.  Iterate this procedure for all possible pair of $i,j$ with $s(e_i)=s(e'_j)$.

We give an example on verifying the sign condition and the non-crossing condition:
\begin{example}\label{eg-walk1}
Let $\mathbb{G}$ be a ribbon graph whose geometric realisation is given on the left-hand side of the following picture.  For readability, we label the half-edges with numeral instead of alphabets and place the labelling next to their respective emanating vertex.
\begin{center}
\begin{picture}(300,120)
\put(0,105){$\xymatrix@C=20pt@R=16pt{
\circ\ar@{-}[rr]^{\overline{2}}\ar@{-}[rrrr]\ar@{-}[rd]^{3}\ar@{-}[rdrd] &&\ar@{-}[rr]^{2}&&\circ\ar@{-}[ldld]\\
&\ar@{-}[rd]^{\overline{3}}&&\ar@{-}[ru]^{\overline{1}}&\\
&&*+[Fo]{v}\ar@{-}[rd]^{\overline{6}}\ar@{-}[rdrd]\ar@{-}[ru]^{1}&&\\
&\ar@{-}[ru]^{4}&&\ar@{-}[rd]^{6}&\\
*+[Fo]{u}\ar@{-}[rr]_{5}\ar@{-}[ruru]\ar@{-}[ru]^{\overline{4}}\ar@{-}[rrrr]&&\ar@{-}[rr]_{\overline{5}}&&\circ
}$}
\put(270,68){\circle*{5}}
\put(270,68){\line(1,1){41}}
\put(220,110){\line(1,0){91}}
\put(220,110){\line(1,-1){110}}
\put(200,0){\line(1,1){60}}
\put(200,0){\line(1,0){130}}
\put(260,60){\line(-1,1){55}}
\put(205,116){\line(1,0){121}}
\put(215,5){\line(1,1){111}}
\put(215,5){\circle*{5}}
\multiput(270,68)(6,0){5}{\line(1,0){3}}
\multiput(215,5)(0,6){5}{\line(0,1){3}}
\end{picture}
\end{center}

Define $w_1$ as the following signed half-walk.
\begin{align}
w_{1}=(1^{+},2^{-},3^{+},\overline{6^{-}},\overline{5^{+}},\overline{4^{-}},\overline{3^{+}},\overline{2^{-}},\overline{1^{+}},4^{-}). \notag
\end{align}
Applying the involution on them we get:
\begin{align}
\overline{w_{1}}=(\overline{4^{-}},1^{+},2^{-},3^{+},4^{-},5^{+},6^{-},\overline{3^{+}},\overline{2^{-}},\overline{1^{+}}).\notag
\end{align}

The walk $W_1$ is shown on the right-hand side of the above picture. We represent the virtual edges as dashed lines to indicate their relative position with respect to the (augmented) cyclic ordering around $u$ and $v$.  From this picture, one then expects that $W_1$ is self-crossing.  We show how this can be checked properly.

We have the following sets:
\begin{align}
w_1\cap w_1 &= \{w_1\},\quad w_1\cap \overline{w_1}= \{z_1=(1,2,3), \overline{z_1}, z_2=(4), \overline{z_2}\},\notag
\end{align}
which gives us the following sets of maximal common subwalks
\begin{align}
W_{1}\cap W_{1}&=\left\{\; W_{1},\; Z_1=\{ z_1, \overline{z_1} \},\; Z_2=\{ z_2,\overline{z_2} \}\; \right\}\notag.
\end{align}
It is easy to see that (NC1) is satisfied in all the cases.  In the above picture, $Z_1$ is represented by the overlapping upper triangle, whereas $Z_2$ is represented by the overlapping edge in the lower triangle.  Focusing on the adjacent half-edges around these overlapping parts, one can see that (NC2) does not hold at both $Z_1$ and $Z_2$.

We will be more precise here.  The neighbourhood cyclic orderings around $s(z_1)$ and $s(\overline{z_1})$ in the half-walk $z_1$ are
\begin{align}
(1,4,\vir_{-}(1))_{v=s(z_1)} \quad \text{and} \quad (\overline{3},4,\overline{6})_{v=s(\overline{z_1})} \quad \text{respectively.}\notag
\end{align}
For $z_2$, we have instead
\begin{align}
(4,1,\overline{3})_{v=s(z_2)}\quad \text{and}\quad (\overline{4},\vir_{+}(\overline{4}),5)_{u=s(\overline{z_2})}. \notag
\end{align}
Therefore, (NC2) does not hold in both cases - as we have claimed.

From the picture above, it is easy to see that $v$ is the only self-intersecting vertex, and we leave it for the reader to confirm the claim.  There are only four neighbourhoods of $v$ in $W_{1}$:
\begin{align}
\{ \vir_{-}(1), 1\}, \quad \{ 1,4\},\quad \{ \overline{3}, 4\}, \quad\text{and }\{ \overline{3}, \overline{6}\}. \notag
\end{align}

Then, to check (NC3), we only need to observe the three cases
$\{ \vir_{-}(1), 1, \overline{3},\overline{6}\}$, $\{ \vir_{-}(1), 1, \overline{3},4\}$, and $\{ 1, 4, \overline{3},\overline{6}\}$.
The associating signed cyclic suborderings are given by
\begin{align}
(\overline{3^{+}},\overline{6^{-}},\; \vir_{-}(1)^{-},1^{+})_{v},\;
(\overline{3^{+}},4^{-},\vir_{-}(1)^{-}, 1^{+},)_{v},\;\text{and }
(1^{+},\overline{3^{+}},4^{-},\overline{6^{-}})_{v} \text{ respectively}. \notag
\end{align}
Thus, while the first two cases comply with (NC3), the third one does not.
\end{example}

\begin{proposition}\label{type-odd}
Let $\mathbb{G}$ be a ribbon graph. Then the following are equivalent.
\begin{itemize}
\item[(1)] $\operatorname{\mathsf{AW}}(\mathbb{G})$ is a finite set.
\item[(2)] $\mathbb{G}$ consists of at most one odd cycle and no even cycle.
\end{itemize}
\end{proposition}

\begin{remark}
A cycle of length $n$ in $\mathbb{G}$ is an $n$-gon embedded in $\mathbb{G}$, i.e. there is a walk $(e_1,\ldots,e_k)$ in $\mathbb{G}$ with endpoints being the same, and no repeating vertices along the walk.  An odd cycle (resp. even cycle) is a cycle of odd (resp. even) length.  Note that a 1-gon (cycle of length 1) is a loop.  We denote a cycle in $\mathbb{G}$ by a sequence $\mathcal{C}=(v,E_1,v_1,E_2,\ldots, v_{k-1},E_k,v)$, so that $s(e_1)=v=s(\overline{e_k})$, $s(e_i)=v_{i-1}=s(\overline{e_{i+1}})$.
\end{remark}

For ease of exposition, regardless of whether the entries of a sequence are half-walk or half-edges, or a mixture of both, the sequence is understood as the half-walk given by concatenating all the data in the obvious way.

\begin{proof}
(2)$\Rightarrow$(1):
Suppose $\mathbb{G}$ is a graph containing at most one odd cycle and no even cycle.
Since sign alternates as we go along a signed walk, the same edge never appears more than once.  In particular, the
length $\ell$ of the sequence defining a signed walk is less than or equal to the number of edges in $\mathbb{G}$.  Moreover, for a given $\ell$, since there is only finitely many sequences (of half-edges) of length $\ell$, $\operatorname{\mathsf{SW}}(\mathbb{G})$ is finite.  Hence, the subset $\operatorname{\mathsf{AW}}(\mathbb{G})$ of $\operatorname{\mathsf{SW}}(\mathbb{G})$ is also finite.

(1)$\Rightarrow$(2):
Assume that $\mathbb{G}$ has an even cycle $\mathcal{C}=(v,E_1,v_1,E_2,\ldots, v_{k-1},E_k,v)$, which has the following geometric realisation in $\mathbb{G}$:
\begin{align}\notag
\xymatrix@R=12pt@C=12pt{\ar@{}[rrdd]|{\mathcal{C}}
v\ar@{-}[dd]_{E_{1}}&&v_{k-1}\ar@{-}[ll]_{E_{k}}\\
&&\\
v_{1}\ar@{--}[rr]&&v_{k-2}\ar@{-}[uu]_{E_{k-1}}
}\end{align}
Note that none of the edges is a loop.  We label the half-edges $e_i$ so that $s(e_i)=v_{i-1}$.  For better readability, we put all the labellings of half-walks and walks in bold face.

Let $\pmb{s}$ be the signed half-walk $(\pmb{e_{1}}^{+},\pmb{e_{2}}^{-},\ldots, \pmb{e_{k}}^{-})$.
For a positive integer $p$, we define a signed half-walk $\pmb{w}_{p}$ and show that its corresponding signed walk $\pmb{W}_p$ is admissible.  For clarity, we will notate some of the vertices (as well as indexing them) along the walk.

Let $\pmb{w}_{p} := (v^{(0)},\pmb{s},v^{(1)},\pmb{s},\ldots,\pmb{s},v^{(p)},\pmb{e_{1}}^{+},v_{1})$ with $v^{(i)}=v$ for each $i\in\{0,1\ldots,p\}$, i.e. concatenation of $p$ copies $\pmb{s}$ and a copy of $\pmb{e_{1}}$.
By construction, there is no self-intersecting vertex in $\pmb{W}_p$.
The set $\pmb{w}_{p}\cap\pmb{w}_{p}$ consists of $\pmb{w}_{p}$ and the following half-walks.
\begin{align}
\begin{array}{rrcl}
\pmb{z}_1: & (v^{(0)},\pmb{e_{1}}^{+},v_{1}) & = & (v^{(p)},\pmb{e_{1}}^{+},v_{1})\\
\pmb{z}_2: & (v^{(0)},\pmb{s},v^{(1)},\pmb{e_{1}}^{+},v_{1}) &=& (v^{(p-1)},\pmb{s},v^{(p)},\pmb{e_{1}}^{+},v_{1})\\
\pmb{z}_3: & (v^{(0)},\pmb{s},v^{(1)},\pmb{s},v^{(2)},\pmb{e_{1}}^{+},v_{1}) &=& (v^{(p-2)},\pmb{s},v^{(p-1)},\pmb{s},v^{(p)},\pmb{e_{1}}^{+},v_1)\\
\vdots & & \vdots &   \\
\pmb{z}_p: & (v^{(0)},\pmb{s},v^{(1)},\pmb{s},\ldots, \pmb{s},v^{(p-1)},\pmb{e_{1}}^{+},v_{1}) &=& (v^{(1)},\pmb{s},v^{(2)},\pmb{s},\ldots,\pmb{s}, v^{(p)},\pmb{e_{1}}^{+},v_{1})
\end{array}\notag
\end{align}
Since the set $\pmb{w}_{p}\cap\overline{\pmb{w}_{p}}$ is empty, $\pmb{W}_{p}\cap\pmb{ W}_{p}=\{\pmb{Z}_1,\ldots,\pmb{Z}_{p},\pmb{W}_{p}\}$.

Now we can see that the sign condition and the non-crossing conditions are satisfied at $\pmb{Z}_i$ for all $i\in\{1,\ldots,p\}$.  Hence, $\pmb{W}_{p}$ is admissible for all $p\geq 1$.  In particular, $\operatorname{\mathsf{AW}}(\mathbb{G})$ is an infinite set.
\medskip

Suppose now that $\mathbb{G}$ has two odd cycles $\mathcal{C}=(u,E_{1},u_{1},E_{2},\ldots,u_{m-1},E_{m},u)$ and $\mathcal{C}'=(v,F_{1},v_{1},F_{2},\ldots,v_{n-1},F_{n},v)$.  First note that we can assume that vertices in $\{u_i,v_j\mid i=1,\ldots,m-1; j=1,\ldots,n-1\}$ are pairwise distinct.  Otherwise, there will be an even cycle subgraph, which reverts us to the previous case, or there is/are odd cycle(s) of shorter length(s) which satisfies the assumption on vertices being pairwise distinction.  Therefore, we have the following two possible geometric realisations:
\begin{align}
\xymatrix@R=12pt{u_1\ar@{--}@/^1pc/[rrdd]& & v_{1}\ar@{--}@/^1pc/[lldd] & & \ar@{}[rdd]|{\mathcal{C}}
u_{1}\ar@{--}@/_/[dd]&&&&&v_{n-1}\\
& v\ar@{-}[lu]^{E_1}\ar@{-}[rd]_{E_m}\ar@{-}[ru]^{F_1}\ar@{-}[ld]_{F_n} & &\text{or} & & u\ar@{-}[lu]_{E_{1}}\ar@{-}[ld]^{E_{m}}\ar@{-}[r]^{D_{1}}&\ar@{.}[r]&\ar@{-}[r]^{D_{l}}&v\ar@{-}[ru]^{F_{n}}\ar@{-}[rd]_{F_{1}}&\\
v_{n-1} &  & u_{m-1} & & u_{m-1}&&&&&v_{1}\ar@{--}@/_/[uu] \ar@{}[luu]|{\mathcal{C}'}
}\notag
\end{align}
where $D_1,\ldots,D_l$ are non-loop edges connecting $\mathcal{C}$ and $\mathcal{C}'$.  Note that $l$ can be zero, i.e. $u=v$.

We label the half-edges as follows.  In the first case, we require the cyclic ordering around $v$ is given by $(\pmb{e_1},\overline{\pmb{f_n}},\overline{\pmb{e_m}},\pmb{f_1})_v$.  In the second case, we require cyclic orderings $(\pmb{e_1},\overline{\pmb{e_m}},\pmb{d_1})_u$ and $(\pmb{f_1},\overline{\pmb{f_n}},\overline{\pmb{d_l}})_v$ if $l>0$, or $(\pmb{e_1},\overline{\pmb{e_m}},\pmb{f_1},\overline{\pmb{f_n}})_v$ if $l=0$.   The remaining labellings are then fixed (uniquely) in the way that the following signed half-walks in $\mathbb{G}$ can be defined:
\begin{align}
\pmb{d}:=(\pmb{d_{1}}^{-},\pmb{d_{2}}^{+},\ldots, \pmb{d_{l-1}}^{-\epsilon},\pmb{d_{l}}^{\epsilon}) \notag \\
\pmb{e}:=(\pmb{e_{1}}^{+},\pmb{e_{2}}^{-},\ldots, \pmb{e_{m-1}}^{-},\pmb{e_{m}}^{+}) \notag \\
\pmb{f}:=(\pmb{f_{1}}^{-\epsilon},\pmb{f_{2}}^{\epsilon},\ldots, \pmb{f_{n-1}}^{\epsilon},\pmb{f_{n}}^{-\epsilon}),\notag
\end{align}
where $\epsilon=+$ in the first case, and $\epsilon$ is the sign of $(-1)^l$ in the second.  Here $\pmb{e}$ and $\pmb{f}$ are half-walks given by walking along respective cycles.

Let us concentrate in the first case for now.  For any positive integer $p$, define the following signed half-walk:
\begin{align}
\pmb{w}_{p}:=(u^{(0)},\pmb{e},u^{(1)},\pmb{f},u^{(2)},\pmb{e},u^{(3)},\pmb{f},u^{(4)},\ldots,\pmb{e},u^{(2p-1)},\pmb{f},u^{(2p)},\pmb{e},u^{(2p+1)}),\notag
\end{align}
where $u^{(i)}=v$ for all $i\in\{0,1,\ldots,2p+1\}$.

Since $v$ is the only vertex appearing in both $\mathcal{C}$ and $\mathcal{C}'$, it is the only self-intersecting vertex of $\pmb{W}_{p}$.
There are only four possible different neighbourhoods of $v$ in $\pmb{W}_{p}$, given by $v$ at $v=u^{(x)}$ with $x\in I:=\{0,1,2,2p+1\}$.
In particular, it makes sense to say ``the neighbourhood of $u^{(x)}$", or ``the neighbourhood at $x$".

Consider $a,b\in I$ with $a<b$.
Now one can simply draw the neighbourhoods (like in the definition of (NC3)) to see that $v$ is a self-intersecting vertex with respect to the neighbourhoods at $a$ and $b$ if, and only if, $b-a$ is odd.
One can simultaneously check that the signed cyclic ordering in (NC3) holds by considering all such $(a,b)\neq(0,2p+1)$.
The case $(a,b)=(0,2p+1)$ gives coinciding endpoints of $\pmb{W}_p$, so we need to check the sign condition, which clearly holds.

To determine maximal common subwalks, (a generalisation of) our previous observation says that we only need to consider $a,b\in\{0,1,\ldots,2p+1\}$ with $a<b$ and $b-a$ even.
Set $\pmb{s}:=(\pmb{e},\pmb{f})$ so that we can write $\pmb{w}_p$ in the form
$(v^{(0)},\pmb{s},v^{(1)},\pmb{s},\ldots,\pmb{s},v^{(p)},\pmb{e},v^{(p+1)})$,
where $v^{(i)}=u^{(2i)}$ for all $i\in\{0,1,\ldots,p\}$, and $v^{(p+1)}=u^{(2p+1)}$.  Then with the new labellings, the maximal (self-)common subwalks obtained by expanding $v^{(a)}=v^{(b)}$ are exactly $\pmb{z}_1,\ldots,\pmb{z}_p,\pmb{w}_p$ after replacing $\pmb{e_1}$ by $\pmb{e}$, and $v_1$ by $v^{(p+1)}$ in the proof for the even cycle case.  These are all the maximal common subwalks and one can check that the corresponding (NC1) and (NC2) conditions hold.
This implies that $\operatorname{\mathsf{AW}}(\mathbb{G})$ is infinite in the first double odd cycle case.

\medskip
Finally, we look at the last case.  We define $\pmb{s}$ as the signed half-walk $(\pmb{d},\pmb{f},\overline{\pmb{d}})$. For every positive integer $p$, we define the following signed half-walk $\pmb{w}_{p}$, and show that it is admissible.
\begin{align}
\pmb{w}_{p}:=& (u^{(0)},\pmb{e},u^{(1)},\pmb{s},u^{(2)},\pmb{e},u^{(3)},\pmb{s},\ldots, \pmb{s},u^{(2p)},\pmb{e},u^{(2p+1)}, \notag \\
& \quad \overline{\pmb{s}}, u^{(2p+2)}, \overline{\pmb{e}}, u^{(2p+3)}, \overline{\pmb{s}}, u^{(2p+4)}, \overline{\pmb{e}}, u^{(2p+5)}, \ldots, \overline{\pmb{s}}, u^{(4p)}, \overline{\pmb{e}}, u^{(4p+1)}  ) , \notag
\end{align}
where $u^{(i)}=u$ for any $i\in \{ 0,1,2,\ldots, 4p+1\}$.

Observe carefully that if one starts with a common vertex not equal to $u$, then one can always expand it to a maximal common subwalk which contains $u$.
Thus, in order to classify maximal common subwalks, and determine the neighbourhoods for which $u$ appears as an intersecting vertex, we start by comparing $u^{(r)}$ with $u^{(s)}$ for some $r,s\in \{ 0,1,\ldots,4p+1 \}$.
A subwalk containing $u=u^{(i)}$ is in one of the following forms:
\begin{center}
\begin{picture}(420,65)(0,0)

\thicklines

\put(15,5){$i\in \{ 0,4p+1\}$}
\put(55,35){\line(-1,1){15}}
\put(55,35){\circle*{4}}
\put(58,25){$u^{(i)}$}
\put(48,45){$e_{1}$}

\put(135,5){$i\in I_{1}\cup I_{2}$}
\put(140,35){\circle*{4}}
\put(140,35){\line(-1,-1){15}}
\put(140,35){\line(1,0){40}}
\put(180,35){\line(1,-1){15}}
\put(137,40){$u^{(i)}$}
\put(135,21){$\overline{e_{m}}$}
\put(175,22){$f_{1}$}

\put(245,5){$i\in J_{1}\cup J_{2}$}
\put(250,35){\circle*{4}}
\put(250,35){\line(-1,1){15}}
\put(250,35){\line(1,0){40}}
\put(290,35){\line(1,1){15}}
\put(247,23){$u^{(i)}$}
\put(243,45){$e_{1}$}
\put(285,44){$\overline{f_{n}}$}

\put(355,5){$i=2p+1$}
\put(360,35){\circle*{4}}
\put(360,35){\line(-1,-1){15}}
\put(360,35){\line(1,0){40}}
\put(400,35){\line(1,1){15}}
\put(355,21){$\overline{e_{m}}$}
\put(395,44){$\overline{f_{n}}$}
\put(357,40){$u^{(i)}$}

\end{picture}
\end{center}
where $I_{1}=\{ 1,3,\ldots, 2p-1\}$, $I_{2}=\{ 2p+2,2p+4,\ldots,4p\}$, $J_{1}=\{ 2,4,\ldots,2p\}$, and $J_{2}=\{ 2p+3, 2p+5,\ldots,4p-1 \}$.

The following list shows all possible intersecting vertices and maximal common subwalks obtained by extending a coinciding indexed vertex $u^{(r)}=u^{(s)}$:
\begin{enumerate}
\item For $r\in \{ 0,4p+1 \}$ and $s\in I_{1}\cup I_{2}$, $u^{(r)}=u^{(s)}$ specifies an intersecting vertex.
\item For $r\in \{ 0,4p+1\}$ and $s=2p+1$, $u^{(r)}=u^{(s)}$ specifies an intersecting vertex.
\item For $r,s\in\{0,4p+1\}$, $u^{(r)}=u^{(s)}$ gives the coinciding endpoints of $\pmb{W}_p$; the induced maximal common subwalk is given by
\begin{align}
(u^{(0)}=u^{(4p+1)},\pmb{e},\ldots,\pmb{s},u^{(2p)}=u^{(2p+1)}) \in \pmb{w}_p\cap \overline{\pmb{w}_p}.\notag
\end{align}
\item For $r=0$ and $s\in J_{1}$, the induced maximal common subwalk is given by
\begin{align}
(u^{(0)}=u^{(s)},\pmb{e},\ldots,\pmb{e},u^{(2p+1-s)}=u^{(2p+1)},\pmb{d}) \in \pmb{w}_p\cap \pmb{w}_p.\notag
\end{align}
Similarly, if $r=4p+1$ instead, then the corresponding maximal common subwalk coming from $\overline{\pmb{w}_p}\cap \pmb{w}_p$ has the same sequence of half-edges.

\item For $r=0$ and $s\in J_{2}$, the induced maximal common subwalk is given by
\begin{align}
(u^{(0)}=u^{(s)},\pmb{e},\ldots,\pmb{s},u^{(s-2p-1)}=u^{(2p+1)}) \in \pmb{w}_p\cap \overline{\pmb{w}_p}.\notag
\end{align}
Similarly, if $r=4p+1$ instead, then the corresponding maximal common subwalk coming from $\overline{\pmb{w}_p}\cap\overline{\pmb{w}_p}$ has the same sequence of half-edges.

\item For $r\in I_{1}$ and $s=2p+1$, the induced maximal common subwalk is given by
\begin{align}
(u^{(0)}=u^{(2p+1-r)},\pmb{e},\ldots,\pmb{e},u^{(r)}=u^{(2p+1)},\pmb{d}) \in \pmb{w}_p\cap \pmb{w}_p.\notag
\end{align}
Similarly, if $r\in I_{2}$ instead, then the corresponding maximal common subwalk coming from $\overline{\pmb{w}_p}\cap \pmb{w}_p$ has the same sequence of half-edges.

\item For $r\in J_{1}$ and $s=2p+1$, the induced maximal common subwalk is given by
\begin{align}
(u^{(0)}=u^{(r+2p+1)},\pmb{e},\ldots,\pmb{s},u^{(r)}=u^{(2p+1)}) \in \pmb{w}_p\cap \overline{\pmb{w}_p}.\notag
\end{align}
Similarly, if $r\in J_{2}$ instead, then the corresponding maximal common subwalk coming from $\overline{\pmb{w}_p}\cap\overline{\pmb{w}_p}$ has the same sequence of half-edges.

\item For $r\in I_{1}$ and $s\in J_{1}$, the induced maximal common subwalk is given by
\begin{align}
(u^{(r)}=u^{(s)}, \pmb{d}) \in \pmb{w}_p\cap \overline{\pmb{w}_p}.\notag
\end{align}
Similarly, if $(r,s)\in I_1\times J_2$ (resp.  $I_{2}\times J_{1}$, resp. $I_{2}\times J_{2}$), then the induced maximal common subwalks are also given by $\pmb{d}\in \pmb{w}_p\cap\pmb{w}_p$ (resp. $\overline{\pmb{w}_p}\cap\overline{\pmb{w}_p}$, resp. $\overline{\pmb{w}_p}\cap\pmb{w}_p$).
Note that if $\pmb{d}$ cannot be defined (i.e. $l=0$), then $u^{(r)}=u^{(s)}$ is an intersecting vertex in all these four cases.

\item For $r,s\in I_{1}$, and assume without loss of generality that $r<s$, the induced maximal common subwalk is given by
\begin{align}
(u^{(0)}=u^{(s-r)},\pmb{e},\ldots,\pmb{e},u^{(r-s+2p+1)}=u^{(2p+1)},\pmb{d}) \in \pmb{w}_p\cap \pmb{w}_p.\notag
\end{align}
Similarly, if $r,s\in J_{1}$, or $(r,s)\in (I_{1}\times I_{2}) \cup (J_{1}\times J_{2})$ with $r+s>4p+1$, then the induced maximal common subwalks are given by the same form of half-walks.

\item For $r,s\in I_{2}$, and assume without loss of generality that $r<s$, the induced maximal common subwalk is given by
\begin{align}
(u^{(2p+1)}=u^{(s-r+2p+1)}, \overline{\pmb{s}},\ldots,\overline{\pmb{e}},u^{(r-s+4p+1)}=u^{(4p+1)}) \in \pmb{w}_p\cap \pmb{w}_p.\notag
\end{align}
Similarly, if $r,s\in J_{2}$, or $(r,s)\in (I_{1}\times I_{2})\cup (J_{1}\times J_{2})$ with $r+s\le 4p+1$, then the induced maximal common subwalks are given by the same form of half-walks.
\end{enumerate}
Using the visualisation above, one can see that the non-crossing conditions are satisfied at each of these cases.  The sign condition is easy to check.  Therefore, $\operatorname{\mathsf{AW}}(\mathbb{G})$ is an infinite set.
\end{proof}

\section{Homological and Algebraic Preliminaries} \label{sec-alg-hom-prelim}
\subsection{Tilting theory}\label{sec-tilting-theory}
We will work in the bounded homotopy category $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ in this subsection.
Without loss of generality, each indecomposable complex takes the form $T=(T^{i},d^{i})_{i\in\mathbb{Z}}$ with $d^i$ lies in the Jacobson radical of $\operatorname{proj}{\Lambda}$ for all $i$.

\begin{definition}
Let $T$ be a complex in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$.
\begin{enumerate}
\item We say that $T$ is \emph{pretilting} if it satisfies $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)}(T,T[n])=0$ for all non-zero integers $n$.
\item We say that $T$ is \emph{tilting} if it is pretilting and generates $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ by taking
direct sums, direct summands, mapping cones and shifts.
\item A pretilting complex is said to be \emph{partial} if it is a direct summand of some tilting complex.
\end{enumerate}
We denote by $\operatorname{\mathsf{tilt}}\Lambda$\label{sym-tilt} the set of isomorphism classes of basic tilting complexes of $\Lambda$.
\end{definition}

For a complex $T$ in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$,
the $n$-th term of $T$ is denoted by $T^n$.
A complex $T$ in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ is called
\begin{itemize}
\item  \emph{stalk} if it is of the form $(0\to T^n\to 0)$ for some $n\in\mathbb{Z}$;
\item \emph{$n$-term} if it is of the form $(0\to T^{-n+1}\to\cdots \to T^{-1}\to T^{0}\to 0)$ for a non-negative integer $n$. Note that this is different from simply requiring $T$ to concentrate on $n$ consecutive degrees.
\end{itemize}

The set of basic tilting complexes up to isomorphism has a natural partial order.

\begin{definitiontheorem}\cite{AI,A}
Let $T$ and $U$ be tilting complexes of $\Lambda$.
We write $T\geq U$\label{sym-TgeqU} if $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)}(T,U[n])=0$ for any positive integer $n$.
Then $\geq$ induces partial order structure on $\operatorname{\mathsf{tilt}}\Lambda$.
Moreover, the set $\ntilt{n}\Lambda$\label{sym-ntilt} of $n$-term tilting complexes is precisely the interval $\{T\in\operatorname{\mathsf{tilt}}\Lambda\mid\Lambda\geq T\geq \Lambda[n-1] \}$ in $\operatorname{\mathsf{tilt}}\Lambda$.
\end{definitiontheorem}

We denote by $\operatorname{\mathsf{2ipt}}\Lambda$\label{sym-ipt} the set of isomorphism classes of indecomposable two-term pretilting complexes of $\Lambda$.
\begin{proposition}\label{prop-noCommonSummand}\label{b}\cite{AH, A}
Let $\Lambda$ be a symmetric algebra and $T$ a two-term pretilting complex of $\Lambda$.
Then the following hold:
\begin{enumerate}[{\rm (i)}]
\item $T$ satisfies $\operatorname{\mathsf{add}} T^0\cap \operatorname{\mathsf{add}} T^{-1}=0$.
\item $T$ is partial tilting.  In fact, it is a direct summand of a two-term tilting complex.
\item It is (two-term) tilting if and only if $|T|=|\Lambda|$.
\end{enumerate}
\end{proposition}

\subsection{Brauer graph algebras and their modules} \label{sec-BGA}
For convenience, we say that a half-edge $e$ is \emph{truncated} if $\sigma(e)=e$ and $\operatorname{\mathfrak{m}}(s(e))=1$.
\begin{definition}\label{def-BGA}
Let $\mathbb{G}$ be a Brauer graph whose multiplicity function is denoted by $\operatorname{\mathfrak{m}}$.

If $\mathbb{G}$ is the connected graph (tree) with one edge $E=\{e,\overline{e}\}$ and two vertices (endpoints of $E$), i.e. $\circ - \circ$, with multiplicity $\operatorname{\mathfrak{m}}\equiv 1$, then we define the Brauer tree algebra of $\mathbb{G}$ as $\Bbbk[\alpha_e]/(\alpha_e^2)$.
Otherwise, we define the Brauer graph algebra $\Lambda_\mathbb{G}$\label{sym-LambdaG} of $\mathbb{G}$ by giving its quiver $Q_\mathbb{G}$\label{sym-QG} and relations as follows.
\begin{itemize}
\item The set of vertices of $Q_\mathbb{G}$ is in bijection with the set of edges in $\mathbb{G}$.  For an edge $E=\{e,\overline{e}\}$ in $\mathbb{G}$, we denote the trivial path corresponding to $E$ by $[e]^0$\label{sym-e0} \emph{and also by} $[\overline{e}]^0$.
\item If $e$ is not truncated, then there is an arrow from $E':=\{\sigma(e),\overline{\sigma(e)}\}$ to $E=\{e,\overline{e}\}$, denoted by $\alpha_{e,\sigma(e)}$\label{sym-a-e-sig-e}.
\end{itemize}

In general, suppose $e':=\sigma^k(e)$ for some $k\leq\mathrm{val}}\newcommand{\vir}{\mathrm{vr}(s(e))$ such that $e$ is not truncated, we define $\alpha_{e,e'}$\label{sym-a-ef} to be the path $\alpha_{e,\sigma(e)}\alpha_{\sigma(e),\sigma^2(e)}\cdots\alpha_{\sigma^{k-1}(e),e'}$.
We call this path \emph{short} if $k\lneqq \mathrm{val}}\newcommand{\vir}{\mathrm{vr}(s(e))$.
Otherwise, the path $\alpha_{e,e}$ will be called a \emph{Brauer cycle}.
For a truncated half-edge $e$, we define the Brauer cycle $\alpha_{e,e}$ to be $\alpha_{\overline{e},\overline{e}}^{\operatorname{\mathfrak{m}}(s(\overline{e}))}$.
For better readability, sometimes we write $(e'|e)$
instead of $\alpha_{e',e}$, and $[e]$\label{sym-a-ee} instead of $\alpha_{e,e}$.
Note that our notation $(e'|e)$ goes in the opposite direction (from right to left) compare to the one in \cite{KS}.

The relations of $\Lambda_\mathbb{G}$ are generated by the following three types of \emph{Brauer relations}.
\begin{itemize}
\item[(Br1)] If both $e$ and $\overline{e}$ are not truncated, then $\alpha_{e,e}^{\operatorname{\mathfrak{m}}(s(e))}-\alpha_{\overline{e},\overline{e}}^{\operatorname{\mathfrak{m}}(s(\overline{e}))}=0$.
\item[(Br2)] $\alpha_{\sigma^{-1}(e),e}\alpha_{\overline{e},\sigma(\overline{e})}=0$ for any $e$ with the indicated arrows in $Q_\mathbb{G}$.
\item[(Br3)] If $\overline{e}$ is truncated, then  $\alpha_{e,e}^{\operatorname{\mathfrak{m}}(s(e))}\alpha_{e,\sigma(e)}=0$.
\end{itemize}
\end{definition}
\begin{remark}
(1) For flexibility, if $E=\{e,\overline{e}\}$ is an edge of $\mathbb{G}$, then the notations $P_E, P_e, P_{\overline{e}}$ all mean the same indecomposable projective module - the one corresponding to $E$.

(2) Recall our convention of taking right modules and identifying maps with arrows of Ext-quiver.  It says that a short path $\alpha_{e',e}$ can be regarded as a map $P_e\to P_{e'}$ given by multiplication of $\alpha_{e',e}$ on the left.  Consequently, we call such a map as \emph{short map}.
\end{remark}

It is well-known that every Brauer graph algebra is a symmetric special biserial algebra and vice versa.
In particular, an indecomposable non-projective module of a Brauer graph algebra falls into either one of the two sub-classes, called string modules and band modules; this is a result of \cite{WW} where these two classes of modules are called representations of the first and second kind respectively.
For reason to be clear later (cf. Lemma \ref{lem-non-zero-hom}), we will not give any more details about band modules, and concentrate only on the string modules.

The usual convention of defining string modules is to use the so-called string combinatorics of special biserial algebras.
In order to avoid using yet another combinatorial theory, we will instead define string modules homologically as follows.
Appendix \ref{subsec-proj-pres} provides more information about string combinatorics.

\begin{definition}\label{def-string-mod}
An indecomposable non-projective $\Lambda_\mathbb{G}$-module $M$ is a string module if its minimal projective presentation $P_M=(P_M^{-1}\xrightarrow{d_M}P_M^0)$\label{sym-PM}\label{sym-dM} can be written in one of the following forms:
\begin{center}
(1) $\xymatrix@R=3pt@C=50pt{
P_{F_1} \ar[rd]|{d_{1,1}} & \\
 &  P_{E_1}\\
P_{F_2} \ar@{.}[dd]\ar[ru]|{d_{1,2}}\ar[rd]|{d_{2,2}}&\\
 & P_{E_2} \ar@{.}[dd]\\
P_{F_{n-1}}\ar[rd]|{d_{m,n-1}} &  \\
 & P_{E_m} \\
P_{F_n} \ar[ru]|{d_{m,n}} &
}$ (2)$\xymatrix@R=3pt@C=50pt{
P_{F_1} \ar[rd]|{d_{1,1}} & \\
 & P_{E_1}\\
P_{F_2} \ar@{.}[ddd]\ar[ru]|{d_{1,2}}\ar[rd]|{d_{2,2}}&\\
 & P_{E_2} \ar@{.}[d]\\
 & P_{E_{m-1}}\\
P_{F_n}\ar[ru]|{d_{m-1,n}}\ar[rd]|{d_{m,n}} &  \\
 & P_{E_m}
}$ (3)$\xymatrix@R=3pt@C=50 pt{
 &  P_{E_1}\\
P_{F_1} \ar[rd]|{d_{2,1}} \ar[ru]|{d_{1,1}} & \\
& P_{E_2} \ar@{.}[dd]\\
P_{F_2} \ar@{.}[dd]\ar[ru]|{d_{2,2}}&\\
 & P_{E_{m-1}}\\
P_{F_n}\ar[ru]|{d_{m-1,n}}\ar[rd]|{d_{m,n}} &  \\
 & P_{E_m} }$
\end{center}
with each $d_{i,j}$ given by a left-multiplication of a path from $F_j$ to $E_i$.
If a complex of projective $\Lambda_\mathbb{G}$-modules takes one of the forms above, or is an indecomposable stalk complex concentrated in degree 0 or $-1$, then it is called a \emph{two-term string complex.}
\end{definition}
\begin{remark}\label{rmk-stringmod}
(i) The definition is presented with a ``chosen direction" to keep it short.  In practice (as well as in our forthcoming proofs), one can reorder $E_i$'s and $F_j$'s ``upside-down", i.e. swap $E_i$ with $E_{m-i+1}$ and swap $F_j$ with $F_{n-j+1}$.  Under such reordering, the diagram of (2) is reflected along the horizontal axis in the middle.

(ii) We can write $d_M$ as a bidiagonal matrix $(d_{i,j})_{i,j}$.  Moreover, if the defining path of $d_{i,j}$ is $[e]^k(e|f)$, then it follows from the Brauer relations that the defining path of the non-zero component $d_{i\pm 1,j}$ (resp. $d_{i,j\pm 1}$) is of the form $[e']^{k'}(e'|\overline{f})$ (resp. $[\overline{e}]^{k''}(\overline{e}|f')$).  (cf. Subsection \ref{subsec-proj-pres})

(iii) We always assume that the minimal projective presentation of a string module takes one of these three forms.
\end{remark}

This is the analogue of complexes which are called homotopy string used in,
for example, \cite{BM} for studying indecomposable objects in the derived category of a gentle algebra.
Since we will only study two-term complexes in this paper, we will drop the adjective ``two-term" for string complexes.

A particular example of string module is a hook module:
\begin{definition}\label{def-hook}
A \emph{hook} (module) is a (string) module of the form $H_{e}:=[e]^0\Lambda/\alpha_{\overline{e},\sigma(\overline{e})}\Lambda$\label{sym-He}, whose minimal projective presentation is given by $P_{\sigma(\overline{e})}\xrightarrow{(\overline{e}\,|\,\sigma(\overline{e}))\cdot-} P_e$.
\end{definition}

\begin{remark}
Let $(e_1,e_2,\ldots,e_k)_v$ be the cyclic ordering around the vertex $v$ in the Brauer graph $\mathbb{G}$.  For each $i\in\{ 1,\ldots,k\}$, the hook module $H_{e_i}$ has the following basis:
\[
\{ [e_i]^l , [e_i]^l(e_i\,\vline\,e_{i+1}),\ldots, [e_i]^l(e_i\,\vline\,e_{i-1}) \mid l=0,\ldots,\operatorname{\mathfrak{m}}(v)-1 \}.
\]
We also use the short hand $\eta_e$\label{sym-hook} the longest path in this basis, i.e. $\eta_e:=[e]^{\operatorname{\mathfrak{m}}(s(e))-1}(e|\sigma^{-1}(e))$ for non-truncated $e$ and $\eta_e:=[e]^0$ otherwise.
The composition factor multiplicity $[H_{e_i}:S_{E_j}]$ is $\operatorname{\mathfrak{m}}(v)$ if $E_j$ is not a loop, or $2\operatorname{\mathfrak{m}}(v)$ otherwise.
Note that if $E=\{e,\overline{e}\}$, then $H_e\ncong H_{\overline{e}}$ unless $\mathbb{G}$ is $\circ -\circ$ with multiplicity $\operatorname{\mathfrak{m}}\equiv 1$ (i.e. $\Lambda_\mathbb{G}=\Bbbk[\alpha_e]/(\alpha_e^2)$).  In particular, our hook is slightly different from the traditional definition \cite{BR}.  Also, if $e$ is a truncated half-edge, then $H_e$ is just the simple module $S_e=S_E$.
\end{remark}

\section{Two-term (pre)tilting complexes via ribbon combinatorics}\label{sec-twotilt}
Let $\mathbb{G}$ be a Brauer graph and $\Lambda:=\Lambda_{\mathbb{G}}$ be the associated Brauer graph algebra.
In this section, we study the relationship between two-term tilting complexes and walks.

Let $T$ be a complex in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$.
If the homology of $T$ is non-trivial only at two consecutive degrees and $T$ is indecomposable, then $T$ is isomorphic to some shift of a two-term complex given by the minimal projective presentation of an indecomposable $\Lambda$-module $H^{n+1}(T)$, and also dually, isomorphic to some shift of the minimal injective (co)presentation of an indecomposable $\Lambda$-module $H^n(T)$.
In particular, we will assume that every indecomposable two-term non-stalk complex in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ takes such a form.

\begin{definition}\label{def-maximal-map}
Suppose $T=(T^{-1}\xrightarrow{d}T^0)$ is a two-term complex in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ such that $T=P_M$ (hence, $d=d_M=(d_{i,j})_{i,j}$) for a string module $M$. Then we call $d$ (resp. $T$, resp. $M$) a \emph{short string} map (resp. complex, resp. module) if every non-zero $d_{i,j}$ is a short map between indecomposable projective modules.
We denote by $\operatorname{\mathsf{2scx}}\Lambda$\label{sym-gcx} the set of indecomposable stalk complexes of projective modules concentrated in degree 0 or $-1$, and two-term short string complexes  $T:=(T^{-1}\xrightarrow{d}T^{0})$ which satisfies $\operatorname{\mathsf{add}} T^{0}\cap \operatorname{\mathsf{add}} T^{-1}=0$.
\end{definition}

For a signed half-walk $w=(e_1,\ldots, e_n;\epsilon)$, define  $T^{0}:=\bigoplus_{\epsilon(e_a)=+}P_{e_{a}}$ and $T^{-1}:=\bigoplus_{\epsilon(e_{b})=-}P_{e_{b}}$.
By the definition of signed half-walks, two consecutive half-edges in $w$ have different signs,
and so we can define a map $T^{-1}\xrightarrow{d} T^0$ with entries $d_{a,b}:P_{e_{b}}\to P_{e_{a}}$ given by
\begin{align}
d_{a,b}:=
\begin{cases}
\alpha_{e_a,\overline{e_{b}}}&\text{if }b=a-1;\\
\alpha_{\overline{e_a},e_b}&\text{if }b=a+1;\\
0&\textnormal{otherwise.}
\end{cases}\notag
\end{align}
We define $T_{w}$\label{sym-Tw} to be the two-term complex given by $(T^{-1}\xrightarrow{d}T^0)$ constructed above.
The following properties are almost immediate from the construction.
\begin{lemma}\label{lem-TW-properties}
\begin{itemize}
\item[(1)] $T_{w}=T_{\overline{w}}$ holds.  In particular, for a signed walk $W=\{w,\overline{w}\}$, we can define $T_W:=T_w=T_{\overline{w}}$\label{sym-TW}.
\item[(2)] $T_{W}$ is in $\operatorname{\mathsf{2scx}}\Lambda$.
\end{itemize}
\end{lemma}
\begin{proof}
(1) This is clear by construction.

(2) This is clear for $w=(e^\pm)$ with $e$ a half-edge.  Otherwise, since we have short maps $d_{a,b}$, the complex $T^{-1}\xrightarrow{d}T^0$ is a minimal projective presentation of the (short string) module $H^0(T)$.
Also, reappearing edges in a signed walk have the same sign, and hence it follows that $\operatorname{\mathsf{add}} T^{-1}\cap\operatorname{\mathsf{add}} T^0=0$.
\end{proof}

\begin{lemma}\label{lem-SW-biject}
The map $\operatorname{\mathsf{SW}}(\mathbb{G})\to\operatorname{\mathsf{2scx}}(\Lambda)$ given by $W\mapsto T_W$ is bijective.
\end{lemma}
\begin{proof}
We prove this by finding the inverse of the map.  We define first a map from $\operatorname{\mathsf{2scx}}(\Lambda)\to \operatorname{\mathsf{SW}}(\mathbb{G})$ given by $T\mapsto W_T=\{w_T,\overline{w_T}\}$.  Note that it is sufficient to define just one half-walk $w_T$ along with signatures on the half-edges in $w_T$.

If $T$ is a stalk complex $P_E[s]$ with $s\in\{0,1\}$, then take $w_T=(e)$ with signature being the sign of $(-1)^s$.
If $T=(\oplus_{j=1}^n P_{F_j} \xrightarrow{d} \oplus_{i=1}^m P_{E_i})$ is not a stalk complex, then it is a minimal projective presentation of a short string module $M=H^0(T)$.
It follows from Remark \ref{rmk-stringmod}(ii) that we can choose the half-edge representatives $e_1,f_1$ of $E_1,F_1$ respectively so that $d_{1,1}=(e_1|\overline{f_1})$ when $M$ is of the form (1) or (2) in Definition \ref{def-string-mod}, or $d=(\overline{e_1}|f_1)$ when $M$ is of the form (3) in Definition \ref{def-string-mod}.
Moreover, these determine canonical representatives $e_i,f_j$ so that $d_{1,2}=(\overline{e_1}|f_2)$, $d_{2,2}=(e_2|\overline{f_2})$, $\ldots$, (resp. $d_{2,1}=(e_2|\overline{f_1})$, $d_{2,2}=(\overline{e_2}|f_2)$, $\ldots$) so on and so forth.
We can now write down a half-walk $(f_1,e_1,f_2,\ldots)$ (resp. $(e_1,f_1,e_2,\ldots)$).
Observe that this half-walk is uniquely defined up to applying the involution $\overline{\,\cdot\,}$.
Since $\operatorname{\mathsf{add}} T^0\cap \operatorname{\mathsf{add}} T^{-1}=0$ and $P_e=P_{\overline{e}}=P_E$, one can define a signature $\epsilon$ on $w_T$ by assigning $\epsilon(e_i)=+$ and $\epsilon(f_j)=-$.

Since a short map $\alpha_{e,\overline{f}}$ corresponds uniquely to an ordered pair $(e,\overline{f})$ of distinct half-edges which are incident to the same vertex, the assignment above gives a well-defined injective map.  Lemma \ref{lem-TW-properties} (2) implies that $W\mapsto T_W$ is the inverse of this map.
\end{proof}

The following says that the bijection given above can be improved to give a combinatorial model which describes the indecomposable two-term pretilting complexes.
\begin{lemma}
Every non-stalk indecomposable two-term pretilting complex $T:=(T^{-1}\xrightarrow{d}T^{0})$ is a short string complex.
In particular, $\operatorname{\mathsf{2ipt}}\Lambda$ is a subset of $\operatorname{\mathsf{2scx}}\Lambda$.
\end{lemma}

\begin{proof}
Suppose $T^{-1}=\bigoplus_{j=1}^n P_{F_j}$ and $T^0=\bigoplus_{i=1}^m P_{E_i}$.
Since $T$ is pretilting,
it follows that $\operatorname{\mathsf{add}} T^0\cap \operatorname{\mathsf{add}} T^{-1}=0$ by Proposition \ref{prop-noCommonSummand}.  In particular, we have $E_i\neq F_j$ for all $i,j$.

Suppose on the contrary that $d_{a,b}:P_{F_b}\to P_{E_a}$ is not short.  As $E_a\neq F_b$, we have a short map $(e_a\,\vline\,\overline{f_b})$.  Let $\alpha:T\to T[1]$ be a map of complexes given by mapping $P_{F_b}$ in $T^{-1}$ to $P_{E_a}$ in $T^0=(T[1])^{-1}$ via $(e_a\,\vline\,\overline{f_b})$.  Non-shortness of $d_{a,b}$ means that we can write it as $[e_a]^k(e_a\,\vline\,\overline{f_b})$ for some $k\geq 1$.  It is easy to see that $\alpha$ is not null-homotopy, or one will have $h\cdot[e_a]^k(e_a\,\vline\,\overline{f_b})=(e_a\,\vline\,\overline{f_b})$ for some $h\in\operatorname{Hom}(P_{e_a},P_{e_a})\cong [e_a]^0\Lambda_\mathbb{G}[e_a]^0$, which is not possible.  We now have a non-zero map $\alpha\in\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G})}(T,T[1])$, contradicting the pretilting-ness of $T$.
\end{proof}

We define one more notion before stating the first main result.
\begin{definition}[Admissible set of signed walks]
A set $\mathbb{W}$ of signed walks is \emph{admissible} if for any pair of (not necessarily distinct) walks $W$ and $W'$ in $\mathbb{W}$, they are non-crossing and satisfy the sign condition.
In particular, admissibility of $W$ is equivalent to that of $\mathbb{W}=\{W\}$.
An admissible set is called \emph{complete} if any admissible set containing $\mathbb{W}$ is $\mathbb{W}$ itself.
Denote by $\operatorname{\mathsf{CW}}(\mathbb{G})$\label{sym-CWG} the set of all complete admissible sets of signed walks.
\end{definition}

For a set $\mathbb{W}$ of signed walks, the map $W\mapsto T_W$ induces a map $\mathbb{W}\mapsto T_\mathbb{W}$, where $T_\mathbb{W}$ is the multiplicity-free (possibly infinite) direct sum of complexes $T_W$ over all $W\in \mathbb{W}$.

\begin{theorem}\label{mainbij}
Let $W,W'$ be signed walks and $T_W, T_{W'}$ be their corresponding two-term complexes (via Lemma \ref{lem-SW-biject}).
Then $T_W\oplus T_{W'}$ is pretilting if and only if $\{W,W'\}$ is admissible.
In particular,
\begin{itemize}
\item $W\mapsto T_W$ defines a bijection $\operatorname{\mathsf{AW}}(\mathbb{G})\to\operatorname{\mathsf{2ipt}}\Lambda_\mathbb{G}$, and
\item $\mathbb{W}\mapsto T_\mathbb{W}$ defines a bijection $\operatorname{\mathsf{CW}}(\mathbb{G})\to\operatorname{\mathsf{2-tilt}}\Lambda_\mathbb{G}$.
\end{itemize}
\end{theorem}

We give a few easy consequences of the theorem before proving it.

\begin{proposition}\label{2tilt}
Let $\mathbb{G}$ be a Brauer graph.
\begin{enumerate}[(1)]
\item If $\mathbb{W}$ is a complete admissible set of signed walks of $\mathbb{G}$ and $\mathbb{G}$ has $n$ edges, then $|W|=n$.

\item Each edge in $\mathbb{G}$ appears in at least one signed walk in a complete admissible set.

\item If $\mathbb{G}'$ is another Brauer graph with the same underlying ribbon graph as $\mathbb{G}$, then there is an isomorphism between the partially ordered sets $\operatorname{\mathsf{2-tilt}}{\Lambda_\mathbb{G}}$ and $\operatorname{\mathsf{2-tilt}}{\Lambda_{\mathbb{G}'}}$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1): By Theorem \ref{mainbij}, $T_\mathbb{W}$ is a tilting complex, which implies $|T_\mathbb{W}|=|\Lambda_\mathbb{G}|=n$.

(2): This follows from the fact that a tilting complex induces a basis of the Grothendieck group $K:=K_0(\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G}))$, and
$K$ is a free abelian group with canonical basis given by the isoclasses of projective indecomposable $\Lambda_\mathbb{G}$-modules, which is in bijection with edges of $\mathbb{G}$.

(3): The set-wise bijection is immediate from Theorem \ref{mainbij}.  The isomorphism on the partial order structure will be shown at the end of Section \ref{sec-proof}.
\end{proof}

\section{Proof of Theorem \ref{mainbij}}\label{sec-proof}
In this section, we give a proof of Theorem \ref{mainbij}.
Let $T:=(T^{-1}\xrightarrow{d}T^{0})$ be a two-term complex in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$.
We define two modules $M_{T}$\label{sym-MT} and $N_{T}$\label{sym-NT} as follows:
\begin{align}
M_{T}:= H^{0}(T),\ N_{T}:=H^{-1}(T)=
\begin{cases}
\Omega^2 M_{T} &(\textnormal{if $T^{0}\neq 0$})\\
T^{-1} &(\textnormal{if $T^{0}=0$})
\end{cases}\notag
\end{align}

The following lemma is the central trick of our proof.
\begin{lemma}\label{lem-non-zero-hom}
Let $\Lambda$ be a symmetric algebra.
Let $T$ and $T'$ be indecomposable two-term complexes in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$.
Then $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)}(T',T[1])=0$ if and only if $\operatorname{Hom}_{\Lambda}(M_{T},N_{T'})=0$.
\end{lemma}
\begin{proof}
Assume that $T'^{0}\neq 0$. Then, by \cite[Lemma 3.4]{AIR},
we have that $\operatorname{Hom}_{\Lambda}(M_{T},\Omega^2 M_{T'})=0$ if and only if $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)}(T',T[1])=0$.
Assume that $T'^{0}=0$. Then we have
\begin{align}
\operatorname{Hom}_{\Lambda}(M_{T}, T'^{-1})=0
&\Leftrightarrow \operatorname{Hom}_{\Lambda}(T'^{-1}, M_{T})=0\notag\\
&\Leftrightarrow \operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)}(T',T[1])=0.\notag
\end{align}
Hence the assertion follows.
\end{proof}

This lemma gives the reason why we are not interested in the band modules.  Indeed, a band module $M$ satisfies $\Omega^2(M)\cong M$, so $\mathrm{id}_M\in \operatorname{Hom}_{\Lambda_\mathbb{G}}(M,\Omega^2 M)=\operatorname{Hom}_{\Lambda_\mathbb{G}}(M,M)$.  In particular, an indecomposable two-term pretilting complex is never isomorphic to a minimal projective presentation of a band module.

Let $T=(T^{-1}\xrightarrow{d} T^{0})$ and $T'=(T'^{-1}\xrightarrow{d'} T'^{0})$ be complexes in $\operatorname{\mathsf{2scx}}\Lambda$.
Let $W:=W_{T}$ and $W':=W_{T'}$ be signed walks with signatures $\epsilon$ and $\epsilon'$ respectively,
corresponding to $T$ and $T'$ under the bijection in Lemma \ref{lem-SW-biject}.
In order to prove Theorem \ref{mainbij}, we need to understand the interaction between the combinatorics of the pair $W,W'$ and the zeroness of $\operatorname{Hom}_\Lambda(M_T,N_{T'})$.
For this purpose, we modify the Loewy diagrams of string modules so that we can find maps from $M_T$ to $N_{T'}$ with ease.

\subsection{Decorated Loewy diagrams}
\label{subsec-ribbon-loewy}

We assume that the reader is familiar with the usage of Loewy diagrams in representation theory, see \cite{Alp,BC} for the abstract theory and \cite{Car} for how the diagrams are used in practice.
For a Brauer tree algebra, Loewy diagrams determine modules uniquely (up to isomorphism) and allow one to use diagrammatic methods for easy homological calculations (cf. Appendix \ref{subsec-appendix-2}).
Unfortunately, Loewy diagrams cannot distinguish different (string) modules for Brauer graph algebras in general.
This can be rectified by using half-edges instead of edges of the defining Brauer graph to label the composition factors.

Note that any uniserial $\Lambda$-module arises from a path $[e]^l(e|\sigma^k(e))$ for some $l,k\geq 0$.
This gives a canonical choice of labelling the composition factors by half-edges.
Namely, the composition factors of such a uniserial module should be labelled, from top to bottom, by $e ,\sigma(e),\sigma^2(e),$ etc.
If we apply this principle to a string module or an indecomposable projective module, then one may need to label a composition factor $S_E$  in the top or socle of the module by both $e$ and $\overline{e}$.
When this happens, we will label such a composition factor by the formal symbol ``$e=\overline{e}$" or ``$\overline{e}=e$".
The choice of which symbol to use depends on where the adjacent composition factors $\sigma^\pm(e)$ and $\sigma^\pm(\overline{e})$ appear; see Example \ref{eg-loewy} below.
We call the resulting diagram a \emph{decorated Loewy diagram}.
Essentially, a decorated Loewy diagram is nothing but a visualisation of the string (in the language of string combinatorics) associated to a string module, cf. Appendix \ref{subsec-colour-quiver}.

\begin{example}\label{eg-loewy}
Let $\mathbb{G}$ be the Brauer graph given by
\[
( V=\{u,v\}, H=\{1,\overline{1},2,\overline{2},3,\overline{3}\}, s, \overline{\,\cdot\,}, \sigma=(1,2,3)_u(\overline{1},\overline{2},\overline{3})_v\; ;\; \operatorname{\mathfrak{m}}\equiv 1).
\]
Its geometric realisation is
\[
\xymatrix@C=40pt{
*+[Fo]{u}\ar@{-}[rr]_{\pmb{1}\qquad\qquad} \ar@/^2pc/@{-}[rr]^{\pmb{3}\qquad\quad} \ar@/^0.8pc/@{-}[r]|{\pmb{2}}& \ar@/_1pc/@{-}[r]&  *+[Fo]{v}
}\]
Here, for $i\in\{1,2,3\}$, we denote by $\pmb{i}$ the genuine edge $\{i,\overline{i}\}$ of $\mathbb{G}$.  Then the classical and decorated Loewy diagram of $P_{\pmb{1}}$ are shown as follows.
\[
\xymatrix@R=5pt@C=5pt{\text{classical:}& & \pmb{1} \ar@{-}[ld] \ar@{-}[rd] &\\
& \pmb{2} \ar@{-}[d]& & \pmb{2}\ar@{-}[d]\\
& \pmb{3} \ar@{-}[rd] & & \pmb{3} \ar@{-}[ld]\\
&& \pmb{1} &
}\qquad
\xymatrix@R=5pt@C=5pt{\text{decorated:}& & 1=\overline{1} \ar@{-}[ld] \ar@{-}[rd] &\\
&2 \ar@{-}[d]& & \overline{2}\ar@{-}[d]\\
&3 \ar@{-}[rd] & & \overline{3} \ar@{-}[ld]\\
&& 1=\overline{1} &
}
\]
In the classical case, one cannot distinguish between the two hook modules $H_1$ and $H_{\overline{1}}$ associated to $P_1$ as they have the same Loewy structure.
The decorated Loewy diagrams of $H_1$ and $H_{\overline{1}}$ are shown as follows.
\[
\xymatrix@R=5pt@C=5pt{H_1: & 1\ar@{-}[d] & & & H_{\overline{1}}: & \overline{1} \ar@{-}[d]\\
&2 \ar@{-}[d]& & & & \overline{2}\ar@{-}[d]\\
&3 & & & & \overline{3}
}
\]
\end{example}

Note that, as one can see from the example above, we always use solid lines to indicate extensions between composition factors.
On the other hand, for simplicity, if the length of a uniserial subquotient is clear from context, then we will only label its top and socle (by half-edges) and join them up by a dotted line.

A decorated Loewy diagram can be drawn by stacking diagrams of other string modules.
The subdiagrams will be labelled by the modules they represent.
We use a solid line to connect appropriate corners of the subdiagrams to indicate where an extension between the representing subquotient occurs - just like how we join up composition factors.
For example, let $e,f$ be some half-edges emanating from the same vertex, and let $M$ be a string module with a uniserial submodule $U$ given by the (short) path $(e|f)$.  Then the decorated Loewy diagram of $M$ is given by
\[
\xymatrix@R=8pt{
e \ar@{.}[rd] & & N\\
 & f=\overline{f} \ar@{-}[ru] &
}
\]
where $N$ represents (the diagram of) $M/U$.
Moreover, it makes sense to say that ``$N$ (in the above diagram) is zero".

In the following, we provide the decorated Loewy diagrams associated to the modules $M=H^0(T_W)$ and $N=H^{-1}(T_W)$ for a signed walk $W$.
We will assume that $W$ is given by $w=(e_1,e_2,\ldots,e_m;\epsilon)$ with $m>1$; otherwise, we have $\{M_T,N_T\}=\{P_{e_1},0\}$ and the diagram for $P_{e_1}$ can be drawn easily as in Example \ref{eg-loewy}.
In the six diagrams below, the framed subdiagrams represent hook modules, and (by the definition of short string modules,) the length of any uniserial subquotient in the unframed parts is always strictly less than the valency of its associating vertex in $\mathbb{G}$.

\underline{Case 1}:  Suppose $W$ is given by $w=(e_1^+,e_2^-,\ldots,e_m^-)$.
Then the decorated Loewy diagrams of $M$ and $N$ are as follows.
\begin{align}
M:\;\;& \xymatrix@C=4pt@R=6pt{
&e_1\ar@{=}[r]&\overline{e_1}\ar@{-}[rd]&&&&&e_{m-1}\ar@{=}[r]&\overline{e_{m-1}}\ar@{.}[rdd]& \\
&&&\sigma(\overline{e_1})\ar@{.}[rdd]&&&\cdots\ar@{.}[ru]&&& \\
&&&&&&\cdots&&&\sigma^{-1}(e_m) \\
\sigma^{-1}(e_1) \ar@{.}[ruuu]&&&&e_2\ar@{=}[r]&\overline{e_2}\ar@{.}[ru]&&&&
\save "1,2"."4,1" *[F]\frm{}
\restore
} \notag \\
& \notag \\
N:\;\;& \hspace*{3cm}\xymatrix@C=4pt@R=6pt{
&&&&e_{m-1}\ar@{=}[r]&\overline{e_{m-1}}\ar@{.}[rdd]&&&& \sigma(\overline{e_m}) \\
\sigma(\overline{e_1})\ar@{.}[rdd]&&&\cdots\ar@{.}[ru]&&&&&& \\
&&&\cdots&&&\sigma^{-1}(e_m)\ar@{-}[rd]&&& \\
&e_2\ar@{=}[r]&\overline{e_2}\ar@{.}[ru]&&&&&e_m\ar@{=}[r]&\overline{e_m}\ar@{.}[ruuu]&
\save "1,10"."4,9" *[F]\frm{}
\restore
}\notag
\end{align}
Note that if $e_1$ is truncated, then the diagram of the hook quotient on the left of $M$ is just a single vertex diagram \fbox{$e_1$}.
Similarly, if $\overline{e_m}$ is truncated, then the diagram of the hook submodule on the right of $N$ is just \fbox{$\overline{e_m}$}.
Similar remarks apply for case 2 and case 3 below.

\underline{Case 2}: Suppose $W$ is given by $w=(e_1^+,e_2^-,\ldots,e_{m}^+)$.
Then the decorated Loewy diagrams of $M$ and $N$ are as follows.
\begin{align}
M:\;\;& \xymatrix@C=4pt@R=6pt{
&e_1\ar@{=}[r]&\overline{e_1}\ar@{-}[rd]&&&&&&&&e_{m}\ar@{=}[r]&\overline{e_{m}}\ar@{.}[rddd]& \\
&&&\sigma(\overline{e_1})\ar@{.}[rdd]&&&\cdots&&&\sigma(e_{m})\ar@{-}[ru] &&& \\
&&&&&&\cdots\ar@{.}[rd]&&&&&& \\
\sigma^{-1}(e_1) \ar@{.}[ruuu]&&&&e_2\ar@{=}[r]&\overline{e_2}\ar@{.}[ru]&&e_{m-1}\ar@{=}[r]&\overline{e_{m-1}}\ar@{.}[ruu]&&&&\sigma^{-1}(\overline{e_{m}})
\save "1,2"."4,1" *[F]\frm{}
\restore
\save "1,12"."4,13" *[F]\frm{}
\restore
}\notag \\
& \notag \\
N:\;\;& \hspace*{3cm}\xymatrix@C=4pt@R=6pt{
\sigma(\overline{e_1})\ar@{.}[rdd]&&&\cdots&&&\sigma(e_{m})\\
&&&\cdots\ar@{.}[rd]&&&\\
&e_2\ar@{=}[r]&\overline{e_2}\ar@{.}[ru]&&e_{m-1}\ar@{=}[r]&\overline{e_{m-1}}\ar@{.}[ruu]&
} \notag
\end{align}

\underline{Case 3}:  Suppose $W$ is given by $w=(e_1^-,e_2^+,\ldots,e_m^-)$.
Then the decorated Loewy diagrams of $M$ and $N$ are as follows.
\begin{align}
M:\;\;& \hspace*{2.5cm} \xymatrix@C=4pt@R=6pt{
&e_2\ar@{=}[r]&\overline{e_2}\ar@{.}[rd]&&e_{m-1}\ar@{=}[r]&\overline{e_{m-1}}\ar@{.}[rdd]& \\
&&&\cdots\ar@{.}[ru]&&& \\
\sigma^{-1}(\overline{e_1}) \ar@{.}[ruu]&&&\cdots&&& \sigma^{-1}(e_m)
} \notag\\
& \notag \\
N:\;\;& \xymatrix@C=4pt@R=6pt{
\sigma(e_1)\ar@{.}[rddd]&&&&e_2\ar@{=}[r]&\overline{e_2}\ar@{.}[rd]&&e_{m-1}\ar@{=}[r]&\overline{e_{m-1}}\ar@{.}[rdd]&&&&\sigma(\overline{e_m}) \\
&&&&&&\cdots\ar@{.}[ru]&&&&&& \\
&&&\sigma^{-1}(\overline{e_1}) \ar@{.}[ruu]&&&\cdots&&& \sigma^{-1}(e_m)\ar@{-}[rd]&&&\\
&e_1\ar@{=}[r]&\overline{e_1}\ar@{-}[ru]&&&&&&&&e_m\ar@{=}[r]&\overline{e_m}\ar@{.}[ruuu]&
\save "1,1"."4,2" *[F]\frm{}
\restore
\save "1,13"."4,12" *[F]\frm{}
\restore
}  \notag
\end{align}

With careful consideration, one can obtain the following observations on the interplay between the decorated Loewy diagram of $M$ (or $N$) and some cyclic suborderings.

\begin{lemma}\label{lem-M-struc}
Let $w=(e_1,e_2,\ldots,e_m;\epsilon)\in W$.
Suppose $\epsilon(e_i)=+$ for some $i\in\{1,2,\ldots,m\}$ and there is $g\neq e_i$ with $s(g)=s(e_i)$.
Then the following are equivalent.
\begin{itemize}
\item[(1)] For some $f\in H$ and (possibly zero) string modules $M_1,M_2$,
the decorated Loewy diagrams of the projective and non-projective $M=H^0(T_W)$ can be drawn respectively as
\begin{align}\label{eq-M-dgm}
\vcenter{\vbox{\xymatrix@C=5pt@R=8pt{
&e_i\ar@{.}[dl]\ar@{=}[r]&\overline{e_i}\ar@{.}[dr]&\\
\ar@{.}[d]&&&\ar@{.}[dd]\\
g\ar@{.}[d]&&&\\
\ar@{.}[dr]&&&\ar@{.}[dl]\\
&e_i\ar@{=}[r] & \overline{e_i} &
\save "1,2"."3,1" *[F]\frm{}
\restore
}}} \qquad \mbox{and} \qquad
\vcenter{\vbox{\xymatrix@C=6pt@R=5pt{
&&&&e_i\ar@{=}[r]&\overline{e_i}\ar@{-}[dr]&\\
&&&&&& M_2\\
&&&g\ar@{.}[uur]&&&\\
M_{1}\ar@{-}[dr] &&&&&& \\
& \overline{f}\ar@{=}[r]&f\ar@{.}[uur]&&&
\save "1,5"."3,4" *[F]\frm{}
\restore}}}
\end{align}
with the property that the framed box is a (uniserial) quotient, denoted by $U$, of $M$.

\item[(1')] (1) holds and $U$ is moreover a short string module.

\item[(2)] $(e_i,g,\overline{e_{i-1}})_{s(e_i)}$ is a cyclic subordering (possibly accounting virtual edges) around $s(e_i)$.
\end{itemize}
\end{lemma}
\begin{proof}
\underline{(1)$\Leftrightarrow$(1')}:  (1') implies (1) is trivial.  Suppose the module $U$ is not short.
Then there is a short string quotient of $U$ with basis $\{[e]^0,\,(e|\sigma(e)),\,\ldots,\,(e|\sigma^k(e))\}$.
We can replace $U$ by this shorter module.

\underline{(1') $\Leftrightarrow$ (2):}
Note that $M=P_{e_i}$ is equivalent to $w=(e_1^+)$.
Therefore, for any $g\neq e_i$, the module $U$ is a quotient of $M$ and there is a cyclic subordering $(e_i,g,\overline{e_{i-1}}=\vir_-(e_i))_{s(e_i)}$ around $s(e_i)$, as required.

In the case when $M$ is non-projective, the diagram (\ref{eq-M-dgm}) means that there is a filtration $M\supset X\supset M_2\supset 0$ with $M/X\cong M_1$.  Let $V$ be the subquotient $X/M_2$.
In particular, the module $U$ defined in (1) and (1') is a quotient of $V$.

Note that $M_1=0$ (i.e. $V$ being a quotient of $M$) is equivalent to $i=1$ or $i=2$.

In the case when $i=1$, we have $\overline{e_{i-1}}=\vir_-(e_i)$. Thus, it is clear from (\ref{eq-cyc-ord-vir}) that (2) holds for any $g\neq e$.
On the other hand, as $V$ is a quotient of $M$, and $i=1$ implies that $V\cong H_e$, we can always find uniserial quotient satisfying (1) or (1').

In the case when $i=2$, we have $\overline{e_{i-1}}=\sigma(f)$.
Since shortness of $M$ is inherited by $V$, every half-edge in the decorated Loewy diagram of $V$ appears at most once.
In particular, for any $g\neq e_i$, we have the cyclic subordering stated in (2).
Conversely, the cyclic subordering given in (2) ensures that the desired quotient $U$ of $V$ exists.

Suppose now $i>2$, or equivalently $M_1\neq 0$.
In this case, we have $\overline{e_{i-1}}=f$.
Similar to the previous paragraph, since $V$ is a short string module, the half-edges in its decorated Loewy diagram are precisely $e_i,\sigma(e_i),\ldots,\sigma^l(e_i)=f$ without repetition.
This means that $U$ is a (short string) quotient if and only if $g=\sigma^k(e_i)$ for some $k<l$, or equivalently $(e_i,g,\overline{e_{i-1}})_{s(e_i)}$ is a cyclic subordering around $s(e_i)$.
\end{proof}

\begin{lemma}\label{lem-N-struc}
Let $w'=(e_1',e_2',\ldots,e_n';\epsilon')\in W'$.
Suppose $\epsilon'(e_j')=-$ for some $j\in\{1,2,\ldots,n\}$ and there is $g'\neq e_j'$ with $s(g')=s(e_j')$.
Then the following are equivalent.
\begin{itemize}
\item[(1)] For some $f'\in H$ and (possibly zero) string modules $N_1,N_2$,
the decorated Loewy diagrams of the projective and non-projective $N=H^{-1}(T_{W'})$ can be drawn respectively as
\begin{align}\label{eq-N-dgm}
\vcenter{\vbox{\xymatrix@C=5pt@R=8pt{
&e_j'\ar@{.}[dl]\ar@{=}[r]&\overline{e_j'}\ar@{.}[dr]&\\
\ar@{.}[d]&&&\ar@{.}[dd]\\
g'\ar@{.}[d]&&&\\
\ar@{.}[dr]&&&\ar@{.}[dl]\\
&e_j'\ar@{=}[r] & \overline{e_j'} &
\save "3,1"."5,2" *[F]\frm{}
\restore
}}} \qquad \mbox{and} \qquad
\vcenter{\vbox{
\xymatrix@C=5pt@R=6pt{
&\overline{f'}\ar@{=}[r]&f'\ar@{.}[ddr]&&&&\\
N_1\ar@{-}[ur]&&&&&& \\
&&&g'\ar@{.}[ddr]&&&\\
&&&&&&N_2\\
&&&&e_j'\ar@{=}[r]&\overline{e_j'}\ar@{-}[ur]&
\save "3,4"."5,5" *[F]\frm{}
\restore}}}
\end{align}
with the property that the framed box is a (uniserial) submodule, denoted by $U'$, of $N$.

\item[(1')] (1) holds and $U'$ is moreover a short string module.

\item[(2)] $(e_j',\overline{e_{j-1}'},g')_{s(e_j')}$ is a cyclic subordering (possibly accounting virtual edges) around $s(e_j')$.
\end{itemize}
\end{lemma}
\begin{proof}
This proof is dual to the previous one.
We explain briefly the ``dual ingredient" needed to prove the equivalence of (1') and (2) in the case when $N$ is non-projective.

The filtration to consider is now $N\supset X\supset N_1\supset 0$ with $N/X\cong N_2$.  Let $V'$ be $X/N_1$.
Similar to the proof of Lemma \ref{lem-M-struc}, there are also three cases (depending solely on $j$) to consider, and they can be summarised as follows.
\begin{align}
\begin{array}{c|ccc}
j & N_1 & V' & \overline{e_{j-1}'} \\
\hline
1 & 0 & H_{\sigma(e_j')} & \vir_+(e_j') \\
2 & 0 & \text{short} & \sigma^{-1}(f') \\
>2 & \text{non-zero} & \text{short} & f'  \\
\end{array}\notag
\end{align}
\end{proof}

\subsection{Analysing Hom-space}
From the decorated Loewy diagrams, one can see that $\operatorname{Hom}_{\Lambda}(M_{T},N_{T'})\neq 0$ if and only if there is a string module $L$ which is isomorphic to both a quotient of $M_{T}$ and a submodule of $N_{T'}$.
For an algebraic argument, one can consult \cite{CB} (cf. Appendix \ref{subsec-appendix-2}).  We thank the referee for pointing us to this reference which we were unaware of.
We will use this diagrammatic technique to analyse the relation between the combinatorics of $W=\{w,\overline{w}\},W'=\{w',\overline{w'}\}$ and the module $L$.

We may assume that $T^{0}\neq 0$ and $T'^{-1}\neq 0$, otherwise $M_T$ and $N_{T'}$ will be zero respectively.

\begin{lemma}\label{lem-L-uniserial}
The following are equivalent:
\begin{itemize}
\item[(U1)] There is a non-zero map $\alpha:M_T\to N_{T'}$ which factors through a uniserial module.

\item[(U2)] There are $w=(e_1,\ldots,e_m)\in W$ and $w'=(e'_1,\ldots,e'_n)\in W'$, $i\in\{1,2,\ldots,m\}$, and $j\in\{1,2,\ldots,n\}$ such that $\epsilon(e_i)=+$, $\epsilon'(e_j')=-$, $s(e_i)=s(e_j')$, and one of the following conditions holds:
\begin{itemize}
\item[(i)] $e_i=e_j'$.
\item[(ii)] $(e_i,e_j',\overline{e_{i-1}}=\overline{e_{j-1}'})_v$ is a cyclic subordering around $v$.
\item[(iii)] $(e_i,e_j',\overline{e_{j-1}'},\overline{e_{i-1}})_v$ is a cyclic subordering around $v$.
\item[(iv)] $(e_i,e_j',\overline{e_{i-1}},\overline{e_{j-1}'})_v$ is a cyclic subordering around $v$.
\end{itemize}
\end{itemize}
\end{lemma}
\begin{proof}
\underline{(U2)$\Rightarrow$(U1)}:
Let $E=\{e_i,\overline{e_i}\}$ and $E'=\{e_j',\overline{e_j'}\}$.
If (U2)(i) holds, then the simple module $S_E$ is (isomorphic to) a direct summand of the top of $M=M_T$ and also a direct summand of the socle of $N=N_{T'}$.  The existence of the map described in (U1) is now clear.

Given (U2) (ii), (iii), or (iv), then we always have the cyclic suborderings $(e_i,e_j',\overline{e_{i-1}})_v$ and $(e_j',\overline{e_{j-1}'},e_i)_v$ around $v$.
We will show that there is a quotient of $M$ and a submodule of $N$ which is isomorphic to the short string uniserial module $L$ with basis $\{[e]^0,(e|\sigma(e)),\ldots,(e|\sigma^k(e))\}$ for some $k<\mathrm{val}}\newcommand{\vir}{\mathrm{vr}(v)$ such that $e=e_i$, $\sigma^k(e)=e_j'$.
Indeed, the existence of the desired quotient follows from Lemma \ref{lem-M-struc} by taking $g=e_j'$.
Dually, we apply Lemma \ref{lem-N-struc} with $g'=e_i$ to obtain the claimed submodule.

\underline{(U1)$\Rightarrow$(U2)}:
Let $U$ be the (uniserial) image of $\alpha$.
Since $U$ is (isomorphic to) a quotient of $M$ and also a submodule of $N$, using an argument similar to the proof of (1)$\Leftrightarrow$(1') in Lemma \ref{lem-M-struc} and Lemma \ref{lem-N-struc}, we can assume that $U$ has basis $\{[e]^0\,(e|\sigma(e)),\ldots,(e|\sigma^k(e))\}$ for some $k\leq \mathrm{val}}\newcommand{\vir}{\mathrm{vr}(s(e))$.

Define $e':=\sigma^k(e)$ and $v:=s(e)$.
The condition implies that $E=\{e,\overline{e}\}$ is in the top of $M$ and $E'=\{e',\overline{e'}\}$ is in the socle of $N$.
Therefore, $e=e_i$ with $\epsilon(e)=+$ for some $i\in\{1,\ldots,m\}$ and $e'=e_j$ with $\epsilon'(e')=-$ for some $j\in\{1,\ldots,n\}$.

We obtain (U2)(i) immediately if $e=e'$.  Otherwise,
we can apply Lemma \ref{lem-M-struc} (resp. Lemma \ref{lem-N-struc}) by taking $g=e_j'$ (resp. $g'=e_i$) to obtain the cyclic subordering  $(e_i,e_j',\overline{e_{i-1}})_v$ (resp. $(e_i,e_j',\overline{e_{j-1}'})_v$) around $v$.
These two cyclic suborderings combine to one of the cases (ii), (iii), or (iv) in (U2).
\end{proof}

\begin{example}\label{eg-loewy1}
Consider the ribbon graph $\mathbb{G}$ and signed half-walk \[
w_1=(1^+,2^-,3^+,\overline{6^-},\overline{5^+},\overline{4^-},\overline{3^+},\overline{2^-},\overline{1^+},4^-)=(e_1^+,e_2^-,\ldots, e_{10}^-)
\]
in Example \ref{eg-walk1}.
Let $\mathbb{G}_0$ be a Brauer graph given by equipping constant multiplicity $\operatorname{\mathfrak{m}}\equiv 1$ to $\mathbb{G}$.
The string modules $M_T$ and $N_T$ have Loewy diagrams:
\[
M_T:\xymatrix@C=2pt@R=2pt{
&&&1\ar@{=}[r]&\overline{1}\ar@{-}[rd]&&&3\ar@{=}[r]&\overline{3}\ar@{-}[rd]&&&&&&&&&&&&&&& \\
&&\overline{3}\ar@{-}[ru]&&&2\ar@{=}[r]&\overline{2}\ar@{-}[ru]&&&4\ar@{-}[rd]&&&\overline{5}\ar@{=}[r]&5\ar@{-}[rd]&&&\overline{3}\ar@{=}[r]&3\ar@{-}[rd]&&&\overline{1}\ar@{=}[r]&1\ar@{-}[rd]&& \\
&4\ar@{-}[ru]&&&&&&&&&\overline{6}\ar@{=}[r]&6\ar@{-}[ru]&&&\overline{4}\ar@{=}[r]&4\ar@{-}[ru]&&&\overline{2}\ar@{=}[r]&2\ar@{-}[ru]&&&\overline{3}& \\
\overline{6}\ar@{-}[ru]&&&&&&&&&&&&&&&&&&&&&&&
\save "1,9"."2,10" *[F]\frm{}
\restore
}
\]
\[
N_T:\xymatrix@C=2pt@R=2pt{
&3\ar@{=}[r]&\overline{3}\ar@{-}[rd]&&&&&&&&&&&&&&&&& \\
\overline{2}\ar@{-}[ru]&&&4\ar@{-}[rd]&&&\overline{5}\ar@{=}[r]&5\ar@{-}[rd]&&&\overline{3}\ar@{=}[r]&3\ar@{-}[rd]&&&\overline{1}\ar@{=}[r]&1\ar@{-}[rd]&&&& \\
&&&&\overline{6}\ar@{=}[r]&6\ar@{-}[ru]&&&\overline{4}\ar@{=}[r]&4\ar@{-}[ru]&&&\overline{2}\ar@{=}[r]&2\ar@{-}[ru]&&&\overline{3}\ar@{-}[rd]&&&5 \\
&&&&&&&&&&&&&&&&&4\ar@{=}[r]& \overline{4}\ar@{-}[ru]&
\save "3,17"."4,18" *[F]\frm{}
\restore
}
\]
The framed parts of the diagrams show that there is a uniserial module which is both a quotient of $M$ and a submodule of $N$.  We can see that there is a cyclic subordering $(\overline{e_3},e_{10},e_4,\overline{e_9})_v$ around $v$ - conforming with (U2)(iii) by taking $w=\overline{w_1}, w'=w_1, i=8, j=10$.
\end{example}

\begin{lemma}\label{lem-L-caseC}
The following are equivalent:
\begin{itemize}
\item[(L1)] There is a non-zero map $\alpha:M_T\to N_{T'}$ which factors through a non-uniserial module.
\item[(L2)] For some $w=(e_1,\ldots,e_m;\epsilon)\in W$,  $w'=(e_1',\ldots,e_n';\epsilon')\in W'$, $i\in\{1,\ldots,m\}$, and $j\in\{1,\ldots,n\}$, there is a signed half-walk $z\in w\cap w'$ given by $z=(t_1,\ldots,t_\ell)$ which satisfies
\begin{itemize}
\item[\textbullet] $t_k=e_{i+k-1}=e_{j+k-1}'$ for all $k\in\{1,\ldots,\ell\}$,
\item[\textbullet] $\epsilon(t_k)=\epsilon'(t_k)$ for all $k\in\{1,\ldots,\ell\}$, and
\item[\textbullet] the neighbourhood cyclic orderings are
$(\overline{e_{j-1}'},\overline{e_{i-1}},t_1)_{s(t_1)}$ and $(\overline{t_\ell},e_{j+\ell}',e_{i+\ell})_{s(\overline{t_\ell})}$.
\end{itemize}
\end{itemize}
The condition (L2) means that the two walks can be visualised locally as the solid lines in the pictures below:
\begin{align}\label{dgm-NC}
\xymatrix@R=20pt@C=20pt{
&\ar@{--}[rd]^{\overline{e_{j-1}'}}&&&&&\\
W:&&u\ar@{-}[r]^{t_1}&\ar@{.}[r]&\ar@{-}[r]^{\overline{t_\ell}}&v\ar@{--}[rd]_{e_{j+\ell}'}\ar@{-}[ru]^{e_{i+\ell}}&\\
&\ar@{-}[ru]_{\overline{e_{i-1}}}&&&&&
}\hspace{5mm}\xymatrix@R=20pt@C=20pt{
&\ar@{-}[rd]^{\overline{e_{j-1}'}}&&&&&\\
W':&&u\ar@{-}[r]^{t_1}&\ar@{.}[r]&\ar@{-}[r]^{\overline{t_\ell}}&v\ar@{-}[rd]_{e_{j+\ell}'}\ar@{--}[ru]^{e_{i+\ell}}&\\
&\ar@{--}[ru]_{\overline{e_{i-1}}}&&&&&
}
\end{align}
where $u=s(t_1)$, $v=s(\overline{t_\ell})$.
\end{lemma}
\begin{proof}
\underline{(L1)$\Rightarrow$(L2):}
Let $L$ be the image of $\alpha$.  Then the decorated Loewy diagram of $L$ is in one of the following three forms.
\begin{align}
\xymatrix@R=5pt@C=7pt{
	&t_1\ar@{=}[r]&\overline{t_1}\ar@{.}[rdd]&&&&&t_{\ell-1}\ar@{=}[r]&\overline{t_{\ell-1}}\ar@{.}[rdd]&&&&\\
	e' \ar@{.}[ru]&&&&&\cdots&\cdots\ar@{.}[ru]&&&&& e \\
	&&&t_2\ar@{=}[r]&\overline{t_2}\ar@{.}[ru]&&&&&t_\ell\ar@{=}[r]&\overline{t_\ell}\ar@{.}[ru]&
}\label{eq-LZ}
\end{align}
\begin{align}
\xymatrix@R=5pt@C=7pt{
	&t_1\ar@{=}[r]&\overline{t_1}\ar@{.}[rdd]&&&&&t_\ell\ar@{=}[r]&\overline{t_\ell}\ar@{.}[rd]&\\
	e' \ar@{.}[ru]&&&&&\cdots&\cdots\ar@{.}[ru]&&& f' \\
	&&&t_2\ar@{=}[r]&\overline{t_2}\ar@{.}[ru]&&&}\label{eq-LM}
\end{align}
\begin{align}
\xymatrix@R=5pt@C=7pt{
	&&&t_2\ar@{=}[r]&\overline{t_2}\ar@{.}[rd]&&&&\\
	f \ar@{.}[rd]&&&&&\cdots&\cdots\ar@{.}[rd]&&& e \\
	&t_1\ar@{=}[r]&\overline{t_1}\ar@{.}[ruu]&&&&&t_\ell\ar@{=}[r]&\overline{t_\ell}\ar@{.}[ru]&
}\label{eq-LW}
\end{align}
where the leftmost uniserial subquotient ending in $t_1$ and the rightmost uniserial subquotient starting in $\overline{t_\ell}$ is non-simple.

By the construction of the decorated Loewy diagrams of $M$ and $N$, we can pick $w=(e_1,\ldots,e_m;\epsilon)\in W$ and $w'=(e_1',\ldots,e_n';\epsilon')\in W'$, $i\in\{1,\ldots,m\}$, and $j\in\{1,\ldots,n\}$ so that $t_k=e_{i+k-1}=e_{j+k-1}'$ for all $k\in\{1,2,\ldots,\ell\}$ with $\epsilon(t_1)=\epsilon'(t_1)$ and $\epsilon(t_\ell)=\epsilon'(t_\ell)$.
Note that
\[
(\epsilon(t_1),\epsilon(t_\ell)) = \begin{cases}
(+,-) & \text{if $L$ takes the form of (\ref{eq-LZ});}\\
(+,+) & \text{if $L$ takes the form of (\ref{eq-LM});}\\
(-,-) & \text{if $L$ takes the form of (\ref{eq-LW}).}
\end{cases}
\]

Since $L$ is a submodule of $N$, we have $e'=\overline{e_{j-1}'}$ and $f'=e_{j+\ell}'$.
Dually, since $L$ is a quotient of $M$, we have $e=e_{i+\ell}$ and $f=\overline{e_{i-1}}$.

Suppose the decorated Loewy diagram of $L$ is in the form of (\ref{eq-LZ}).
Since $e'=\overline{e_{j-1}'}$, we have $e'\neq t_1= e_i$.  Then we can apply Lemma \ref{lem-M-struc} on $w$ with $g=\overline{e_{j-1}'}$ to obtain the cyclic subordering $(\overline{e_{j-1}'},\overline{e_{i-1}},e_i)_{s(t_1)}$.
Moreover, we now have $\overline{e_{i-1}}\neq \overline{e_{j-1}'}$, which means that $t_1$ is the starting term of some $z\in w\cap w'$.
Likewise, $e=e_{i+\ell}$ means that we can apply Lemma \ref{lem-N-struc} on $\overline{w'}$ with $g'=e_{i+\ell}$ to obtain the cyclic subordering $(\overline{e_{j+\ell-1}'},e_{j+\ell}',e_{i+\ell})_{s(\overline{t_\ell})}$ and $e_{i+\ell}\neq e_{j+\ell}'$.
This means that the maximal common subwalk given by $z$ ends with $t_\ell=e_{i+\ell-1}=e_{j+\ell-1}'$.
Note that the two cyclic suborderings obtained above are precisely the neighbourhood cyclic orderings of $z$, which are of the form stated in (L2).

Similarly, to obtain (L2) in the case of (\ref{eq-LM}), we apply Lemma \ref{lem-M-struc} on $w$ with $g=\overline{e_{j-1}'}$ and on $\overline{w}$ with $g=e_{j+\ell}'$, whereas in the case of (\ref{eq-LW}), we apply Lemma \ref{lem-N-struc} on $w'$ with $g'=\overline{e_{i-1}}$ and on $\overline{w'}$ with $g'=e_{i+\ell}$.

\underline{(L2)$\Rightarrow$(L1):}
We want to write down the decorated Loewy diagrams so that we can find a quotient $L$ of $M$ which is isomorphic to a submodule of $N$, or vice versa.

If $\epsilon(t_1)=+$ and $\epsilon(t_\ell)=-$, then by the given form of the neighbourhood cyclic orderings, $\overline{e_{j-1}'}$ and $e_{i+\ell}$ cannot be virtual.
Now we can apply Lemma \ref{lem-M-struc} on $w$ with $g=\overline{e_{j-1}'}$, and Lemma \ref{lem-N-struc} on $\overline{w'}$ with $g'=e_{i+\ell}$, which gives the following decorated Loewy diagrams for $M$ and $N$:
\begin{align}
& \xymatrix@R=5pt@C=5pt{
&&&&{\phantom{\overline{e_{j-1}'}}}&e_i\ar@{=}[r]&\overline{e_i}\ar@{.}[rdd]&&&&&&&&\\
M:&M_{1}\ar@{-}[rd]&&& \overline{e_{j-1}'} \ar@{.}[ru]&&&&&\cdots\ar@{.}[rd]&&& e_{i+\ell}\ar@{=}[r]&\overline{e_{i+\ell}}\ar@{.}[rd]&\\
&&\overline{f}\ar@{=}[r]&f\ar@{.}[ru]&&&&e_{i+1}\ar@{=}[r]&\overline{e_{i+1}}\ar@{-}[ru]&&e_{i+\ell-1}\ar@{=}[r]&\overline{e_{i+\ell-1}}\ar@{.}[ru]&{\phantom{g_{i+1}}}&&M_{2}
\save "1,5"."3,13" *[F]\frm{}
\restore
}\notag \\
& \xymatrix@R=5pt@C=5pt{
&N_1\ar@{-}[rd]&&{\phantom{\overline{e_{j-1}'}}}&e_j'\ar@{=}[r]&\overline{e_j'}\ar@{.}[rdd]&&&&&&&e_{j+\ell}'\ar@{=}[r]&\overline{e_{j+\ell}'}\ar@{-}[rd]&\\
N:&&e_{j-1}'\ar@{=}[r]&\overline{e_{j-1}'}\ar@{.}[ru]&&&&&\cdots\ar@{.}[rd]&&&e_{i+\ell}\ar@{.}[ru]&&&N_2\\
&&&&&&e_{j+1}'\ar@{=}[r]&\overline{e_{j+1}'}\ar@{.}[ru]&&e_{j+\ell-1}'\ar@{=}[r]&\overline{e_{j+\ell-1}'}\ar@{.}[ru]&{\phantom{\overline{g_{i+\ell}}}}&&&
\save "1,4"."3,12" *[F]\frm{}
\restore
}\notag
\end{align}
where the framed parts of the diagrams are both isomorphic to a module $L$ of the form (\ref{eq-LZ}) with $e'=\overline{e_{j-1}'}$ and $e=e_{i+\ell}$.
We can see from the diagrams that $L$ is a quotient of $M$ and a submodule of $N$, and so (L1) holds.

Similarly, if $\epsilon(t_1)$ and $\epsilon(t_\ell)$ are both positive (resp. both negative), then we can apply Lemma \ref{lem-M-struc} on $w$ and on $\overline{w}$ (resp. Lemma \ref{lem-N-struc} on $w'$ and on $\overline{w'}$) with an appropriate choice of indices to construct the decorated Loewy diagrams of $M$ and $N$, so that the desired $L$ is in the form (\ref{eq-LM}) (resp. (\ref{eq-LW})).
\end{proof}

\begin{example}\label{eg-loewy2}
We consider the Brauer graph and the walk $w_1$ used in Example \ref{eg-loewy1}.
There is a maximal common subwalk given by $z_2=(4^-)\in w_1\cap \overline{w_1}$ which looks locally in $\mathbb{G}$ as follows (cf. figures in Example \ref{eg-walk1}):
\[
\xymatrix@R=5pt{
\ar@{--}[rdddd]^<{\vir_+(\overline{4})}& & & &\ar@{-}[ldd]_<{\overline{3}}\\
& & & &\\
& \ar@{-}[rr]^{\overline{4} \qquad 4} & & &\\
& *+[Fo]{u} & & *+[Fo]{v}& \\
& \ar@{-}[rr] & & &\\
& & & & \\
\ar@{-}[ruuuu]^<{5}& & & & \ar@{-}[luu]_<{1}
}
\]
where we use the dashed line to represent virtual edge as in Example \ref{eg-walk1}.
Taking $w=\overline{w_1}$, $w'=w_1$, and $z=z_2$, we see that the condition of (L2) is met.

In the following decorated Loewy diagrams, the boxes in solid lines  indicate the module $L$ (of the form (\ref{eq-LZ})) in the previous lemma.
Note that we have ``flipped" the diagram of $N_T$ (when compared to the one in Example \ref{eg-loewy1}) as the common subwalk given by $z_1$ comes from comparing $w_1$ and $\overline{w_1}$.
\[
M_T:\xymatrix@C=2pt@R=2pt{
&&&1\ar@{=}[r]&\overline{1}\ar@{-}[rd]&&&3\ar@{=}[r]&\overline{3}\ar@{-}[rd]& {\vphantom{1}}&&&&&&&&&&&&&& \\
&&\overline{3}\ar@{-}[ru]&&&2\ar@{=}[r]&\overline{2}\ar@{-}[ru]&&&4\ar@{-}[rd]&&&\overline{5}\ar@{=}[r]&5\ar@{-}[rd]&&&\overline{3}\ar@{=}[r]&3\ar@{-}[rd]&&&\overline{1}\ar@{=}[r]&1\ar@{-}[rd]&& \\
&4\ar@{-}[ru]&&&&&&&&&\overline{6}\ar@{=}[r]&6\ar@{-}[ru]&&&\overline{4}\ar@{=}[r]&4\ar@{-}[ru]&&&\overline{2}\ar@{=}[r]&2\ar@{-}[ru]&&&\overline{3}& \\
\overline{6}\ar@{-}[ru]&&&&&&&&&&&&&&&&&&&&&&&
\save "1,14"."3,17" *[F]\frm{}
\restore
\save "1,10"."3,2" *[F--]\frm{}
\restore
}
\]
\[
N_T:\xymatrix@C=2pt@R=2pt{
&&&&&&&&&&&&&&&&&\overline{3}\ar@{=}[r]&3\ar@{-}[rd]&& \\
&&{\vphantom{1}}&&1\ar@{=}[r]&\overline{1}\ar@{-}[rd]&&&3\ar@{=}[r]&\overline{3}\ar@{-}[rd]&{\vphantom{1}}&&5\ar@{=}[r]&\overline{5}\ar@{-}[rd]&&&4\ar@{-}[ru]&&&\overline{2} \\
5\ar@{-}[rd]&&&\overline{3}\ar@{-}[ru]&&&2\ar@{=}[r]&\overline{2}\ar@{-}[ru]&&&4\ar@{=}[r]&\overline{4}\ar@{-}[ru]&&&6\ar@{=}[r]&\overline{6}\ar@{-}[ru]&&&&& \\
&\overline{4}\ar@{=}[r]&4\ar@{-}[ru]&&&&&&&&&&&&&&&&&&
\save "3,1"."4,4" *[F]\frm{}
\restore
\save "2,3"."4,11" *[F--]\frm{}
\restore}
\]

For the other maximal subwalk given by $z_1=(1^+,2^-,3^+)\in w_1\cap\overline{w_1}$, we can visualise the local structure around the endpoints as follows:
\[
\xymatrix@R=5pt@C=30pt{
\ar@{-}[rdddd]^<{4}& & & & &\ar@{-}[ldd]_<{\overline{3}}\\
& & & & &\\
& \ar@{-}[r]^{\overline{3}} & \ar@{.}[r]& & \ar@{-}[l]_{1} &\\
& *+[Fo]{v} & & & *+[Fo]{v}& \\
& \ar@{-}[r] & \ar@{.}[r]& &\ar@{-}[l]&\\
& & & & \\
\ar@{-}[ruuuu]^<{\overline{6}}& & & & & \ar@{--}[luu]_<{\vir_-(1)}
}
\]
This indicates that $z_1$ fits the criteria for (L2); the corresponding module $L$ (in the form of (\ref{eq-LM})) is shown in the Loewy diagrams above framed by the boxes in dashed line.
\end{example}

The following proposition is the final piece needed to prove Theorem \ref{mainbij}.
\begin{proposition}\label{lem-hom-to-comb}
\begin{itemize}
\item[(1)] Retaining the notations used so far, $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G})}(T',T[1])=0$ if and only if neither (U2) nor (L2) holds for $W=W_{T}$ and $W'=W_{T'}$.
\item[(2)] For two (not necessarily distinct) complexes $U,V\in \operatorname{\mathsf{2scx}}\Lambda_\mathbb{G}$, the hom-spaces $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G})}(U,V[1])$ and $\operatorname{Hom}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G})}(V,U[1])$ are simultaneously zero if, and only if $W_{U},W_{V}$ are non-crossing and satisfy the sign condition.
\end{itemize}
\end{proposition}
Note that the right-hand side of (2) is not equivalent to saying $\{W_U,W_V\}$ is admissible, as it does not require $W_U$ and $W_V$ to be admissible.
\begin{proof}
(1) This follows from combining Lemma \ref{lem-non-zero-hom}, \ref{lem-L-uniserial}, and \ref{lem-L-caseC}.

(2) First, we clarify in the following table the relation between the non-crossing and signs conditions and the conditions in (U2), (L2).
In each row of the table, the condition in the first entry fails to hold for $\{W_U, W_V\}$ implies that one of the conditions in the second entry holds for $(W,W')=(W_U,W_V)$ or $(W,W')=(W_V, W_U)$.
For example, if $\{W_U,W_V\}$ fails (NC1), then at least one of the pairs $(W_U,W_V)$ or $(W_V,W_U)$ satisfies one of the conditions (U2)(i) or (U2)(ii).
On the other hand, if a condition appears on the right-hand side of the table, then one of the corresponding left-hand side conditions will fail.
For example, say (U2)(i) holds for $(W_U,W_V)$, then $\{W_U,W_V\}$ fails (NC1) or the sign condition.
\begin{center}
\begin{tabular}{c|ll}
$\begin{array}{c} \text{Condition which fails}\\ \text{for }\{W_U,W_V\} \end{array}$  &  &  Condition which holds for $(W_U,W_V)$ or $(W_V,W_U)$ \\
\hline
(NC1) & & (U2)(i) or (ii) \\
(NC2) & & (L2) \\
(NC3) & & (U2)(iii) or (iv) \\
sign condition & & (U2)(i) or (iii)
\end{tabular}
\end{center}
So now the statement follows by applying (1) to $(T,T')=(U,V)$ and to $(T,T')=(V,U)$ simultaneously.
\end{proof}

\begin{proof}[Proof of Theorem \ref{mainbij}]
Suppose $U,V$ are (possibly the same) complexes in $\operatorname{\mathsf{2scx}}\Lambda_\mathbb{G}$.
By definition, $U\oplus V$ being pretilting means that the four hom-spaces $\operatorname{Hom}(U,U[1])$, $\operatorname{Hom}(U,V[1])$, $\operatorname{Hom}(V,V[1])$, $\operatorname{Hom}(V,U[1])$ are all zero. By Proposition \ref{lem-hom-to-comb}, this is equivalent to $\{W_U,W_V\}$ being an admissible set.
This implies that a (basic) two-term pretilting complex correspond to precisely an admissible set of signed walks.  The first bijection in the statement of the theorem is just the special case when $U=V$.
More generally, we get that $T_\mathbb{W}$ is pretilting for any admissible set $\mathbb{W}$ of signed walks.

Since any two-term pretilting complex $T$ is partial, if $T\oplus U$ is a two-term pretilting complex with $T$ tilting, then $U\in \operatorname{\mathsf{add}} T$.
Translating this to the combinatorial side, we obtain that $T_\mathbb{W}$ is tilting precisely when $\mathbb{W}$ is complete.
\end{proof}

\begin{proof}[Proof of Proposition \ref{2tilt}]
Note that the set $\operatorname{\mathsf{AW}}(\mathbb{G})$, and hence $\operatorname{\mathsf{CW}}(\mathbb{G})$, does not depend on the multiplicity of $\mathbb{G}$,
that is, we have natural bijections $\operatorname{\mathsf{AW}}(\mathbb{G})\to\operatorname{\mathsf{AW}}(\mathbb{G}')$ and $\operatorname{\mathsf{CW}}(\mathbb{G})\to\operatorname{\mathsf{CW}}(\mathbb{G}')$.
These yield bijections $\psi:\operatorname{\mathsf{2ipt}}\Lambda_\mathbb{G}\to\operatorname{\mathsf{2ipt}}\Lambda_{\mathbb{G}'}$ and
(by abusing notation) $\psi:\operatorname{\mathsf{2-tilt}}\Lambda_\mathbb{G}\to\operatorname{\mathsf{2-tilt}}\Lambda_{\mathbb{G}'}$ by Theorem \ref{mainbij}.

We are left to show that $\psi$ preserves the partial order.
Let $T,T'\in\operatorname{\mathsf{2-tilt}}\Lambda_\mathbb{G}$ with $T\geq T'$.
This means that $\operatorname{Hom}(X,X'[1])=0$ for all indecomposable summands $X$ and $X'$ of $T$ and $T'$ respectively.
Let $W_X$ be the signed walk corresponding to an indecomposable summand $X$ of $T$.
Recall from the construction (cf. Lemma \ref{lem-SW-biject}) that $W_X$ is independent of the multiplicities on the vertices.
Hence, the indecomposable summand $\psi(X)$  of $\psi(T)$ (resp. $\psi(X')$ of $\psi(T')$) corresponds also to $W_X$ (resp. $W_{X'}$).
Therefore, as Proposition \ref{lem-hom-to-comb} (1) is independent of multiplicity, we have $\operatorname{Hom}(\psi(X), \psi(X')[1])=0$; hence, $\psi(T)\geq \psi(T')$, i.e. $\psi$ is order-preserving.
\end{proof}

\section{Tilting-discrete Brauer graph algebras}\label{sec-application}
\subsection{Preliminaries}\label{sec-application-prelim}
We first recall some results about tilting mutation theory from \cite{AI,A,AIR}.
Throughout this subsection, $\Lambda$ is assumed to be a finite dimensional \emph{symmetric} algebra.  Most of the facts stated here have analogues for general finite dimensional algebras by replacing the word ``tilting" with ``silting"; see \emph{loc. cit.} for the details.

Let $\mathcal{A}$ be a full additive subcategory of $\mathcal{C}=\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ or $\mod\Lambda$.
A map $f:X\to Y$ in $\mathcal{C}$ is \emph{left minimal} if all maps $g:Y\to Y$ with $gf=f$ are isomorphisms.
A map $f:U\to X$ in $\mathcal{C}$ is called a \emph{left $\mathcal{A}$-approximation} of $U$
if $X$ belongs to $\mathcal{A}$ and $\operatorname{Hom}_{\mathcal{C}}(f,C)$ is surjective for any $C$ in $\mathcal{A}$.
We say that $\mathcal{A}$ is \emph{covariantly finite} in $\mathcal{C}$ if
for all $U$ in $\mathcal{C}$, there exists a left $\mathcal{A}$-approximation.
Dually, we define right minimality, right $\mathcal{A}$-approximations, and contravariantly finite subcategories in $\mathcal{C}$.
A full subcategory is called \emph{functorially finite} if it is covariantly and contravariantly finite in $\mathcal{C}$.

\begin{definitiontheorem}\cite{AI}
Let $T$ be a basic tilting complex in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)$ with a decomposition $T=X\oplus M$.
A \emph{left tilting mutation} $\mu_X^-(T)$\label{sym-muXT} of $T$ with respect to $X$ is a (basic) tilting complex given by $Y\oplus M$, where $Y$ is the (well-defined) object fitting into the following triangle:
\[\xymatrix{
X \ar[r]^f & M' \ar[r] & Y \ar[r] & X[1]
}\]
where $f$ is a minimal left $\operatorname{\mathsf{add}} M$-approximation of $X$.
Dually, one also has another tilting complex given by \emph{right tilting mutation} $\mu_X^+(T)$ of $T$ with respect to $X$.

A \emph{tilting mutation} means a left or right tilting mutation; it is called \emph{irreducible} if $X$ is indecomposable.
\end{definitiontheorem}

\begin{definition}\cite{AI}
The \emph{tilting quiver} of $\Lambda$, denoted by $\mathcal{T}_\Lambda$\label{sym-TLambda}, is defined as follows:
\begin{itemize}
\item The set of vertices is $\operatorname{\mathsf{tilt}}\Lambda$.
\item Draw an arrow $T\to U$ if $U$ is an irreducible left mutation of $T$.
\end{itemize}  Note that this is precisely the Hasse quiver of the poset $(\operatorname{\mathsf{tilt}}\Lambda,\geq)$.
A connected component of $\mathcal{T}_\Lambda$ is said to be \emph{canonical} if it contains $\Lambda$.
\end{definition}

\begin{definition}
A symmetric algebra $\Lambda$ is said to be \emph{tilting-connected} if
the tilting quiver of $\Lambda$ is connected.
We say that $\Lambda$ is \emph{tilting-discrete} if for any positive integer $\ell$,
there exist only finitely many tilting complexes $T$ in $\operatorname{\mathsf{tilt}}\Lambda$ satisfying $\Lambda\geq T\geq\Lambda[\ell]$.
\end{definition}
It was shown in \cite[Section 3]{A} that if $\Lambda$ is tilting-discrete, then it is tilting-connected.

Two classes of tilting-discrete symmetric algebras are
found in the second author's previous works.  Namely, the local  algebras in \cite{AI} and the representation-finite algebras in \cite{A}.

We refer to \cite{AI} for more general examples of tilting/silting-connected algebras.
On the other hand, in a joint work of the second author with Grant and Iyama \cite{AGI}, we know that there exist non-tilting-connected symmetric algebras.
In any case, it is not easy to answer the following question.

\begin{question}
When is a symmetric algebra tilting-connected, or even tilting-discrete?
\end{question}

The next subsection is devoted to answering the tilting-discrete part of the question for Brauer graph algebras.

We first look at some properties when the set $\operatorname{\mathsf{2-tilt}}\Lambda$ is finite, or equivalently (by Proposition \ref{prop-noCommonSummand} (ii)), when the set $\operatorname{\mathsf{2ipt}}\Lambda$ is finite.

\begin{proposition}\cite{A}\label{prop-locTiltDiscreteMutate}
If $\operatorname{\mathsf{2-tilt}}\Lambda$ is a finite set,
then any two-term tilting complex can be obtained from $\Lambda$ by iterated irreducible left tilting mutation.
\end{proposition}

The following is a special case of \cite[Theorem 2.4]{AM}.

\begin{proposition}\cite{AM}
\label{tilting-discrete}
Let $\Lambda$ be a symmetric algebra.
If for any tilting complex $P$ in the canonical connected component of $\mathcal{T}_\Lambda$, the set
$\operatorname{\mathsf{2-tilt}}\operatorname{End}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda)}(P)$ is finite,
then $\Lambda$ is tilting-discrete.
In particular, it is tilting-connected.
\end{proposition}

\subsection{Conditions for tilting-discreteness}\label{sec-tilt-discrete}
The aim of this subsection is to show the following theorem.

\begin{theorem}\label{TD of type odd}
Let $\mathbb{G}$ be a Brauer graph.
Then the following are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $\Lambda_\mathbb{G}$ is tilting-discrete;
\item $\operatorname{\mathsf{2-tilt}}\Lambda_\mathbb{G}$ is a finite set;
\item $\mathbb{G}$ contains at most one odd cycle and no even cycle.
\end{enumerate}
\end{theorem}

First of all, we start by recalling the Brauer graph mutation theory from \cite{Kau,A2}.
\begin{definition}[Flips of Brauer graph] \label{def-flip}
Let $\mathbb{G}=(V,H,s,\overline{\,\cdot\,},\sigma;\operatorname{\mathfrak{m}})$ be a Brauer graph and $E$ an edge of $\mathbb{G}$.  If $\mathbb{G}$ has more than one edge, then the \emph{left flip of $\mathbb{G}$ at $E$} is a Brauer graph $\mu^-_E(\mathbb{G})$\label{sym-muEG} given by $(V,H,s',\overline{\,\cdot\,},\sigma';\operatorname{\mathfrak{m}})$, where $s'$ and $\sigma'$ are defined as follows.
\[
\begin{array}{c|l|c|c}
\text{half-edge $h$} & \multicolumn{1}{|c|}{\text{condition}} & s'(h) & \sigma'(h) \\ [0.2em] \hline
\hline
e \in E & \sigma^{-1}(e)\notin E;\;\; \sigma(e)\neq \overline{e}    & s\left(\overline{\sigma^{-1}(e)}\right) & \overline{\sigma^{-1}(e)} \rule{0pt}{1.5em} \\ [0.6em]
 & \sigma^{-1}(e)\notin E;\;\; \sigma(e)=\overline{e} & s\left(\overline{\sigma^{-1}(e)}\right) & \overline{e}   \\[0.6em]
 & \sigma^{-1}(e) = e     & s(e)   & e        \\[0.6em]
 & \sigma^{-1}(e) =\overline{e} & s\left(\overline{\sigma^{-1}(\overline{e})}\right)=s'(\overline{e})   & \overline{\sigma^{-1}(\overline{e})}                     \\[0.6em]
\hline
f\notin E & f=\sigma^{-1}\left(\overline{\sigma^{-1}(e)}\right)& s(f) & e \rule{0pt}{1.5em} \\ [0.6em]
          & f=\sigma^{-1}(e);\;\; \sigma(e)=\overline{f}        & s(f) & e \\ [0.6em]
          & f=\sigma^{-1}(e);\;\; \sigma(e)=\overline{e};\;\;\; \sigma(\overline{e})=\overline{f}     & s(f) & e \\ [0.6em]
          & f=\sigma^{-1}(e);\;\; \sigma(e)=\overline{e};\;\;\; \sigma(\overline{e})\neq\overline{f}  & s(f) & \sigma^2(e) \\ [0.6em]
          & f=\sigma^{-1}(e);\;\; \sigma(e)\neq\overline{e},\overline{f}     & s(f) & \sigma(e)  \\ [0.6em]
          & \text{all other cases}          & s(f) & \sigma(f) \\ [0.6em]
\end{array}
\]
If $\mathbb{G}$ has one edge, then the left flip of $\mathbb{G}$ is defined to be itself.

The \emph{opposite (Brauer) graph} of $\mathbb{G}$ is the Brauer graph $\mathbb{G}^\mathsf{op} = (V,H,s,\overline{\,\cdot\,},\sigma^{-1})$\label{sym-Gop}.  The \emph{right flip of $\mathbb{G}$ at $E$} is the Brauer graph $\mu_E^+(\mathbb{G}):=\mu_E^-(\mathbb{G}^\mathsf{op})^\mathsf{op}$.
\end{definition}

The simplest way to present a graphical presentation of the left flip is given below.
If $E$ is not a loop, i.e. $u,v$ in the graphs below are distinct vertices:
\[
\xymatrix@C=35pt{
a \ar@{-}[dr]_{}="au"& & & c \ar@{-}[dl]^{}="cu" & a \ar@{-}[dr]_{}="au2"& & & c \ar@{-}[dl]^{}="cu2"\\
  & u \ar@{-}[r]^{E} &  v&  \ar[r]^{\mu_E^-}&   & u  &  v&  \\
b \ar@{-}[ur]^{}="bu"& & & d \ar@{-}[ul]_{}="du" & b \ar@{-}[uurrr]^{E}\ar@{-}[ur]^{}="bu2"& & & d \ar@{-}[ul]_{}="du2"
\ar@{.}@/_/ "au";"bu"
\ar@{.}@/^/ "cu";"du"
\ar@{.}@/_/ "au2";"bu2"
\ar@{.}@/^/ "cu2";"du2"
}
\]
Here we allow some (or all) of $a,b,v$ to be the same vertex; similarly for $c,d,u$.

If $E$ is a loop, then we have one of the following two cases.
\begin{center}
\begin{picture}(250,120)(0,-10)
\put(0,7.5){{$v$}}
\put(-50,7.5){{$a$}}
\put(50,7.5){{$b$}}
\put(-43,10){\line(1,0){42}}
\put(7,10){\line(1,0){40}}
\put(-52,13){\line(-1,1){20}}
\put(-52,7){\line(-1,-1){20}}
\put(56,13){\line(1,1){20}}
\put(56,7){\line(1,-1){20}}
\qbezier(0,15)(-18,30)(-20,10)
\qbezier(-20,10)(-18,-10)(0,5)
\put(-20,25){{$E$}}

\put(90,10){\vector(1,0){50}}
\put(110,20){{$\mu_E^-$}}

\put(220,7.5){{$v$}}
\put(170,7.5){{$a$}}
\put(270,7.5){{$b$}}
\put(177,10){\line(1,0){42}}
\put(227,10){\line(1,0){40}}
\put(168,13){\line(-1,1){20}}
\put(168,7){\line(-1,-1){20}}
\put(276,13){\line(1,1){20}}
\put(276,7){\line(1,-1){20}}
\qbezier(265,15)(210,30)(200,10)
\qbezier(200,10)(190,-10)(175,5)
\put(210,25){{$E$}}

\put(0,57.5){{$u$}}
\put(50,57.5){{$v$}}
\put(7,60){\line(1,0){40}}
\qbezier(0,65)(-20,85)(3,90)
\qbezier(3,90)(26,85)(6,65)
\put(-23,75){{$E$}}

\put(90,60){\vector(1,0){50}}
\put(110,70){{$\mu_E^-$}}

\put(170,57.5){{$u$}}
\put(220,57.5){{$v$}}
\put(177,60){\line(1,0){40}}
\qbezier(220,65)(200,85)(223,90)
\qbezier(223,90)(246,85)(226,65)
\put(240,75){{$E$}}
\end{picture}
\end{center}
Here $a,b$ can be the same vertex. cf. \cite[Section 5]{A2}
\begin{remark}
\begin{enumerate}[(1)]
\item We will abuse the notation $\mu_E^\pm(\mathbb{G})$ to mean ``$\mu_E^-(\mathbb{G})$, and respectively $\mu_E^+(\mathbb{G})$," for ease of stating results.  Similar abuses of notations will also be adopted for mutations of tilting complexes.
\item Intuitively, if one draws $\mathbb{G}$ on a piece of paper and places a mirror perpendicular to the paper next to $\mathbb{G}$, then $\mathbb{G}^\mathsf{op}$ is the reflection of $\mathbb{G}$ in the mirror.
\item The opposite ring $(\Lambda_\mathbb{G})^\mathsf{op}$ of $\Lambda_\mathbb{G}$ is isomorphic to $\Lambda_{\mathbb{G}^\mathsf{op}}$.
\item One can also define the right flip explicitly, then left flip will be given by $\mu_E^-(\mathbb{G})=\mu_E^+(\mathbb{G}^\mathsf{op})^\mathsf{op}$.
\item The left/right flip at $E$ for non-loop $E$ was found by Kauer \cite{Kau}, and is termed as Kauer move in \cite{MS}.  The terminology ``flip" was adopted in \cite{A} to align with the flip of triangulations in cluster mutation theory.  Indeed, if the underlying ribbon graph of $\mathbb{G}$ is a triangulation of a Riemann surface, then $\mu_E^\pm(\mathbb{G})$ is precisely the flip of the triangulation at the arc $E$.
\end{enumerate}
\end{remark}

\begin{proposition}\label{prop-flip der equiv}
Let $\mathbb{G}$ be a Brauer graph and $E$ an edge of $\mathbb{G}$.  Then the endomorphism ring of the tilting complex $\mu_{P_E}^\pm(\Lambda_\mathbb{G})$ is isomorphic to $\Lambda_{\mu_E^\pm(\mathbb{G})}$.
\end{proposition}
\begin{remark}\label{rmk-der-eq}
This was claimed in \cite{Kau,An2,A2}.
All these proofs rely on a claim from \cite{Pog}, namely, that any algebra stably equivalent to a self-injective special biserial algebra is also special biserial.
However, the proof of this claim is incorrect \cite[Appendix]{AIP}; the validity of the claim is still unknown at the time of writing.
However, we were informed by Alexandra Zvonareva that  these proofs can be fixed by \cite{AZ2}, where they prove that the class of Brauer graph algebras is closed under derived equivalences.
We will fix the gap in the proof from \cite{A2} in Appendix \ref{sec-app-endo}; Demonent also gives an explicit calculation of this claim in \cite{Dem}, similar to Appendix \ref{sec-app-endo}, for a more generalised class of algebras.
\end{remark}

\begin{lemma}\label{flip and type odd}
Let $\mathbb{G}$ be a Brauer graph and $E$ an edge of $\mathbb{G}$.
If $\mathbb{G}$ contains $c$ odd cycles with $c\in\{0,1\}$, and no even cycle, then so does $\mu_E^\pm(\mathbb{G})$.
\end{lemma}
\begin{proof}
First note that $\mu_E^\pm(\mathbb{G})$ preserves connectedness.  In particular, if $\mathbb{G}$ is a (connected) tree (i.e. $c=0$), or equivalently $|V|=|H/\overline{\,\cdot\,}|+1$, then so is $\mu_E^\pm(\mathbb{G})$.

Assume now that $\mathbb{G}$ has exactly one odd cycle and has no even cycle.  Now we have $|V|=|H/\overline{\,\cdot\,}|$, and by the same argument, $\mu_E^\pm(\mathbb{G})$ must then contain precisely one cycle.  We are left to show that the parity of the cycle length remains unchanged after a flip.

Since flipping an edge does not alter the rest of the graph, the odd cycle of $\mathbb{G}$ stays as the same subgraph if we flip at an edge which is not in the cycle.  If $E$ is contained in the odd cycle, say of length $\ell$, of $\mathbb{G}$, then observe using the graphical presentation of flips that the cycle length can only be $\ell$, or $\ell-2$, or $\ell+2$.
\end{proof}

Now we are ready to prove Theorem \ref{TD of type odd}.
\begin{proof}[Proof of Theorem \ref{TD of type odd}]
It is evident that the implication (i)$\Rightarrow$(ii) holds.

The implications (ii)$\Leftrightarrow$(iii) follows from Theorem \ref{mainbij} and Proposition \ref{type-odd}.

We show that the implication (iii)$\Rightarrow$(i) holds.
Let $P$ be a tilting complex of $\Lambda_\mathbb{G}$ in the canonical connected component of $\mathcal{T}_{\Lambda_\mathbb{G}}$.
Then it follows from Proposition \ref{prop-flip der equiv} that the endomorphism algebra $\Gamma:=\operatorname{End}_{\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G})}(P)$ of $P$ is a Brauer graph algebra
whose Brauer graph $\mathbb{G}'$ is obtained by a series of left and right flips starting from $\mathbb{G}$.
By Lemma \ref{flip and type odd}, $\mathbb{G}'$ then has the same number (zero or one) of odd cycle as $\mathbb{G}$, and it also has no even cycle.  Combining Theorem \ref{mainbij} and Proposition \ref{type-odd}, we get that $\operatorname{\mathsf{2-tilt}}\Gamma$ is a finite set.  Hence, $\Lambda_\mathbb{G}$ is tilting-discrete by Proposition \ref{tilting-discrete}.
\end{proof}

The following corollary is immediate from (the proof of) Theorem \ref{TD of type odd}.

\begin{corollary}\label{cor-type-odd-equiv}
Let $\mathbb{G}$ be a Brauer graph which contains at most one odd cycle and no even cycle, and let $T$ be a tilting complex of $\Lambda_\mathbb{G}$.
Then the endomorphism algebra of $T$ in $\mathsf{K}^{\rm b}(\operatorname{proj}\Lambda_\mathbb{G})$ is isomorphic to $\Lambda_{\mathbb{G}'}$ for some $\mathbb{G}'$ with the same number of odd cycles, no even cycle, and the same multiplicity as $\mathbb{G}$.  In particular, any algebra derived equivalent to $\Lambda_\mathbb{G}$ is also a Brauer graph algebra.
\end{corollary}

The class of algebras described in Theorem \ref{TD of type odd} appears in \cite{An1} as (precisely) the class of Brauer graph algebras with non-degenerate Cartan matrices.  Note that the Grothendieck group of the sable module category of such a Brauer graph algebra is finite, and vice versa.

For an arbitrary symmetric algebra $\Lambda$, we do not know if the non-degeneracy of its Cartan matrix, or the finiteness of the set $\operatorname{\mathsf{2-tilt}}\Lambda$, is an equivalent condition for tilting-discreteness.  We also do not know if the finiteness, or even if the number of elements, of the set $\operatorname{\mathsf{2-tilt}}\Lambda$ is derived invariant.  In fact, one of the original motivations of this work and its sequel is to see if one can take advantages of the rich combinatorics of Brauer trees in order to count the number of two-term tilting complexes for Brauer tree algebras.

Suppose $\mathbb{G}$ is a Brauer graph such that $\Lambda_\mathbb{G}$ is tilting-discrete.
Let $\mathbb{G}_0$ be the associated \emph{multiplicity-free Brauer graph}, i.e. the Brauer graph with the same ribbon graph structure but multiplicity $\operatorname{\mathfrak{m}}\equiv 1$.
Then the associated Brauer graph algebra $\Lambda_{\mathbb{G}_0}$ is of finite type when $c=0$, or of one-parametric Euclidean type when $c=1$.  See for example \cite{Sko} for the details.

Also note that the multiplicity-free Brauer graph algebras forms precisely the class of trivial extensions of gentle algebras \cite{Sch}.  For readers who are familiar with silting theory \cite{AI} and silting-discreteness \cite{A}, the result in \cite[Proposition 6.8]{BPP} asserts that all derived-discrete algebras of finite global dimension are silting-discrete.  However, the trivial extensions of these algebras also contains multiplicity-free Brauer graph algebras which lie outside the class presented in Theorem \ref{TD of type odd}.  In particular, this shows that silting-discreteness is one of the many properties destroyed by taking trivial extension.

Although Theorem \ref{TD of type odd} gives the precise condition for a Brauer graph algebra to be tilting-discrete,
we still do now know if there is a tilting-connected non-tilting-discrete Brauer graph algebra.
We remark that, in another on-going work of the second named author with Grant and Iyama \cite{AGI},
the Brauer graph algebra whose underlying graph is a digon (i.e. cycle of length 2) is neither tilting-discrete nor tilting-connected.

Our final remark to this result is the problem of classifying derived equivalent classes of algebras.  Such classification for Brauer graph algebras is known in the case where the Brauer graph having at most one odd cycle, no even cycle, and at most two vertices with multiplicity higher than 1, see the survey \cite{Sko} for details.  So Corollary \ref{cor-type-odd-equiv} is a slight generalisation of known results but with an entirely different approach.

redpajama_set_name: RedPajamaArXiv