Disconnected cuts in 4-connected planar graphs
Let $G=(V,E)$ be a connected graph. A subset $S\subset V$ is a cut of $G$ if $G-S$ is disconnected. A near triangulation is a 2-connected plane graph that has at most one face that is not a triangle. In this paper, we explore minimal cuts of 4-connected planar graphs. Our main result is that every minimal cut of a 4-connected planar graph $G$ is connected if and only if $G$ is a near-triangulation. We use this result to sketch a linear-time algorithm for finding a disconnected cut of a 4-connected planar graph.