Graph Subset Problem

Description:

Input:

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 $$$2 \cdot 10^5$$$.

Output:

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.

Sample Input:

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

Sample Output:

2
4 1 2 3 
1 10
1 2 3 4 5 6 7 8 9 10 
-1

Note:

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 $$$-1$$$.