A13951. 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 $-1$ .