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Contest: Task: Related: TaskE TaskG

Score : $500$ points

Problem Statement

You are given an undirected graph $G$ with $N$ vertices and $M$ edges. $G$ is simple (it has no self-loops and multiple edges) and connected.

For $i = 1, 2, \ldots, M$, the $i$-th edge is an undirected edge $\lbrace u_i, v_i \rbrace$ connecting Vertices $u_i$ and $v_i$.

Construct two spanning trees $T_1$ and $T_2$ of $G$ that satisfy both of the two conditions below. ($T_1$ and $T_2$ do not necessarily have to be different spanning trees.)

$T_1$ satisfies the following.

If we regard $T_1$ as a rooted tree rooted at Vertex $1$, for any edge $\lbrace u, v \rbrace$ of $G$ not contained in $T_1$, one of $u$ and $v$ is an ancestor of the other in $T_1$.
$T_2$ satisfies the following.

If we regard $T_2$ as a rooted tree rooted at Vertex $1$, there is no edge $\lbrace u, v \rbrace$ of $G$ not contained in $T_2$ such that one of $u$ and $v$ is an ancestor of the other in $T_2$.

We can show that there always exists $T_1$ and $T_2$ that satisfy the conditions above.

Constraints

$2 \leq N \leq 2 \times 10^5$
$N-1 \leq M \leq \min\lbrace 2 \times 10^5, N(N-1)/2 \rbrace$
$1 \leq u_i, v_i \leq N$
All values in input are integers.
The given graph is simple and connected.

Input

Input is given from Standard Input in the following format:

$N$ $M$
$u_1$ $v_1$
$u_2$ $v_2$
$\vdots$
$u_M$ $v_M$

Output

Print $(2N-2)$ lines to output $T_1$ and $T_2$ in the following format. Specifically,

The $1$-st through $(N-1)$-th lines should contain the $(N-1)$ undirected edges $\lbrace x_1, y_1\rbrace, \lbrace x_2, y_2\rbrace, \ldots, \lbrace x_{N-1}, y_{N-1}\rbrace$ contained in $T_1$, one edge in each line.
The $N$-th through $(2N-2)$-th lines should contain the $(N-1)$ undirected edges $\lbrace z_1, w_1\rbrace, \lbrace z_2, w_2\rbrace, \ldots, \lbrace z_{N-1}, w_{N-1}\rbrace$ contained in $T_2$, one edge in each line.

You may print edges in each spanning tree in any order. Also, you may print the endpoints of each edge in any order.

$x_1$ $y_1$
$x_2$ $y_2$
$\vdots$
$x_{N-1}$ $y_{N-1}$
$z_1$ $w_1$
$z_2$ $w_2$
$\vdots$
$z_{N-1}$ $w_{N-1}$

Sample Input 1

Sample Output 1

In the Sample Output above, $T_1$ is a spanning tree of $G$ containing $5$ edges $\lbrace 1, 4 \rbrace, \lbrace 4, 3 \rbrace, \lbrace 5, 3 \rbrace, \lbrace 4, 2 \rbrace, \lbrace 6, 2 \rbrace$. This $T_1$ satisfies the condition in the Problem Statement. Indeed, for each edge of $G$ not contained in $T_1$:

for edge $\lbrace 5, 1 \rbrace$, Vertex $1$ is an ancestor of $5$;
for edge $\lbrace 1, 2 \rbrace$, Vertex $1$ is an ancestor of $2$;
for edge $\lbrace 1, 6 \rbrace$, Vertex $1$ is an ancestor of $6$.

$T_2$ is another spanning tree of $G$ containing $5$ edges $\lbrace 1, 5 \rbrace, \lbrace 5, 3 \rbrace, \lbrace 1, 4 \rbrace, \lbrace 2, 1 \rbrace, \lbrace 1, 6 \rbrace$. This $T_2$ satisfies the condition in the Problem Statement. Indeed, for each edge of $G$ not contained in $T_2$:

for edge $\lbrace 4, 3 \rbrace$, Vertex $4$ is not an ancestor of Vertex $3$, and vice versa;
for edge $\lbrace 2, 6 \rbrace$, Vertex $2$ is not an ancestor of Vertex $6$, and vice versa;
for edge $\lbrace 4, 2 \rbrace$, Vertex $4$ is not an ancestor of Vertex $2$, and vice versa.

Sample Input 2

Sample Output 2

Tree $T$, containing $3$ edges $\lbrace 1, 2\rbrace, \lbrace 1, 3 \rbrace, \lbrace 1, 4 \rbrace$, is the only spanning tree of $G$. Since there are no edges of $G$ that are not contained in $T$, obviously this $T$ satisfies the conditions for both $T_1$ and $T_2$.