CF1439B Graph Subset Problem

You 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.

The 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 $ .

For 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.

In 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 $ .