A14889 | Neighbour Ordering

Given an undirected graph $G$ , we say that a neighbour ordering is an ordered list of all the neighbours of a vertex for each of the vertices of $G$ . Consider a given neighbour ordering of $G$ and three vertices $u$ , $v$ and $w$ , such that $v$ is a neighbor of $u$ and $w$ . We write $u <_{v} w$ if $u$ comes after $w$ in $v$ 's neighbor list.

A neighbour ordering is said to be good if, for each simple cycle $v_1, v_2, \ldots, v_c$ of the graph, one of the following is satisfied:

- $v_1 <_{v_2} v_3, v_2 <_{v_3} v_4, \ldots, v_{c-2} <_{v_{c-1}} v_c, v_{c-1} <_{v_c} v_1, v_c <_{v_1} v_2$ .
- $v_1 >_{v_2} v_3, v_2 >_{v_3} v_4, \ldots, v_{c-2} >_{v_{c-1}} v_c, v_{c-1} >_{v_c} v_1, v_c >_{v_1} v_2$ .

Given a graph $G$ , determine whether there exists a good neighbour ordering for it and construct one if it does.

The input consists of multiple test cases. The first line contains a single integer $t$ ( $1 \leq t \leq 10^4$ ) — the number of test cases. Description of the test cases follows.

The first line of each test case contains two integers $n$ and $m$ ( $2 \leq n \leq 3 \cdot 10^5$ , $1 \leq m \leq 3 \cdot 10^5$ ), the number of vertices and the number of edges of the graph.

The next $m$ lines each contain two integers $u, v$ ( $0 \leq u, v < n$ ), denoting that there is an edge connecting vertices $u$ and $v$ . It is guaranteed that the graph is connected and there are no loops or multiple edges between the same vertices.

The sum of $n$ and the sum of $m$ for all test cases are at most $3 \cdot 10^5$ .

For each test case, output one line with YES if there is a good neighbour ordering, otherwise output one line with NO. You can print each letter in any case (upper or lower).

If the answer is YES, additionally output $n$ lines describing a good neighbour ordering. In the $i$ -th line, output the neighbours of vertex $i$ in order.

If there are multiple good neigbour orderings, print any.