CF1133F2 Spanning Tree with One Fixed Degree

You are given an undirected unweighted connected graph consisting of $ n $ vertices and $ m $ edges. It is guaranteed that there are no self-loops or multiple edges in the given graph. Your task is to find any spanning tree of this graph such that the degree of the first vertex (vertex with label $ 1 $ on it) is equal to $ D $ (or say that there are no such spanning trees). Recall that the degree of a vertex is the number of edges incident to it.

The first line contains three integers $ n $ , $ m $ and $ D $ ( $ 2 \le n \le 2 \cdot 10^5 $ , $ n - 1 \le m \le min(2 \cdot 10^5, \frac{n(n-1)}{2}), 1 \le D < n $ ) — the number of vertices, the number of edges and required degree of the first vertex, respectively. The following $ m $ lines denote edges: edge $ i $ is represented by a pair of integers $ v_i $ , $ u_i $ ( $ 1 \le v_i, u_i \le n $ , $ u_i \ne v_i $ ), which are the indices of vertices connected by the edge. There are no loops or multiple edges in the given graph, i. e. for each pair ( $ v_i, u_i $ ) there are no other pairs ( $ v_i, u_i $ ) or ( $ u_i, v_i $ ) in the list of edges, and for each pair $ (v_i, u_i) $ the condition $ v_i \ne u_i $ is satisfied.

If there is no spanning tree satisfying the condition from the problem statement, print "NO" in the first line. Otherwise print "YES" in the first line and then print $ n-1 $ lines describing the edges of a spanning tree such that the degree of the first vertex (vertex with label $ 1 $ on it) is equal to $ D $ . Make sure that the edges of the printed spanning tree form some subset of the input edges (order doesn't matter and edge $ (v, u) $ is considered the same as the edge $ (u, v) $ ). If there are multiple possible answers, print any of them.

The picture corresponding to the first and second examples:  The picture corresponding to the third example: