A15143. Lenient Vertex Cover

You are given a simple connected undirected graph, consisting of $n$ vertices and $m$ edges. The vertices are numbered from $1$ to $n$ .

A vertex cover of a graph is a set of vertices such that each edge has at least one of its endpoints in the set.

Let's call a lenient vertex cover such a vertex cover that at most one edge in it has both endpoints in the set.

Find a lenient vertex cover of a graph or report that there is none. If there are multiple answers, then print any of them.

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of testcases.

The first line of each testcase contains two integers $n$ and $m$ ( $2 \le n \le 10^6$ ; $n - 1 \le m \le \min(10^6, \frac{n \cdot (n - 1)}{2})$ ) — the number of vertices and the number of edges of the graph.

Each of the next $m$ lines contains two integers $v$ and $u$ ( $1 \le v, u \le n$ ; $v \neq u$ ) — the descriptions of the edges.

For each testcase, the graph is connected and doesn't have multiple edges. The sum of $n$ over all testcases doesn't exceed $10^6$ . The sum of $m$ over all testcases doesn't exceed $10^6$ .

For each testcase, the first line should contain YES if a lenient vertex cover exists, and NO otherwise. If it exists, the second line should contain a binary string $s$ of length $n$ , where $s_i = 1$ means that vertex $i$ is in the vertex cover, and $s_i = 0$ means that vertex $i$ isn't.

If there are multiple answers, then print any of them.

Here are the graphs from the first example. The vertices in the lenient vertex covers are marked red.