B. Graph Subset Problemtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given an undirected graph with n vertices and m edges. Also, you are given an integer k.Find either a clique of size k or a non-empty subset of vertices such that each vertex of this subset has at least k neighbors in the subset. If there are no such cliques and subsets report about it.A subset of vertices is called a clique of size k if its size is k and there exists an edge between every two vertices from the subset. A vertex is called a neighbor of the other vertex if there exists an edge between them.InputThe first line contains a single integer t (1 \leq t \leq 10^5) — the number of test cases. The next lines contain descriptions of test cases.The first line of the description of each test case contains three integers n, m, k (1 \leq n, m, k \leq 10^5, k \leq n).Each of the next m lines contains two integers u, v (1 \leq u, v \leq n, u \neq v), denoting an edge between vertices u and v.It is guaranteed that there are no self-loops or multiple edges. It is guaranteed that the sum of n for all test cases and the sum of m for all test cases does not exceed 2 \cdot 10^5.OutputFor each test case: If you found a subset of vertices such that each vertex of this subset has at least k neighbors in the subset in the first line output 1 and the size of the subset. On the second line output the vertices of the subset in any order.If you found a clique of size k then in the first line output 2 and in the second line output the vertices of the clique in any order.If there are no required subsets and cliques print -1.If there exists multiple possible answers you can print any of them.ExampleInput
3
5 9 4
1 2
1 3
1 4
1 5
2 3
2 4
2 5
3 4
3 5
10 15 3
1 2
2 3
3 4
4 5
5 1
1 7
2 8
3 9
4 10
5 6
7 10
10 8
8 6
6 9
9 7
4 5 4
1 2
2 3
3 4
4 1
1 3
Output
2
4 1 2 3 
1 10
1 2 3 4 5 6 7 8 9 10 
-1
NoteIn the first test case: the subset \{1, 2, 3, 4\} is a clique of size 4.In the second test case: degree of each vertex in the original graph is at least 3. So the set of all vertices is a correct answer.In the third test case: there are no cliques of size 4 or required subsets, so the answer is -1.
3
5 9 4
1 2
1 3
1 4
1 5
2 3
2 4
2 5
3 4
3 5
10 15 3
1 2
2 3
3 4
4 5
5 1
1 7
2 8
3 9
4 10
5 6
7 10
10 8
8 6
6 9
9 7
4 5 4
1 2
2 3
3 4
4 1
1 3

2 4 1 2 3  1 10 1 2 3 4 5 6 7 8 9 10  -1 
['constructive algorithms', 'data structures', 'graphs', '*2600']