Score : $500$ points

问题描述

我们有一个简单的无向图 $G$，包含 $(S+T)$ 个顶点和 $M$ 条边。顶点编号从 $1$ 到 $(S+T)$，边编号从 $1$ 到 $M$。第 $i$ 条边连接顶点 $u_i$ 和 $v_i$。

这里，顶点集合 $V_1 = \lbrace 1, 2,\dots, S\rbrace$ 和 $V_2 = \lbrace S+1, S+2, \dots, S+T \rbrace$ 都是独立集。

长度为 $4$ 的环称为 4-环。
如果 $G$ 中包含一个 4-环，请任意选择其中一个，并按照任意顺序打印该环中的顶点。
如果 $G$ 不包含 4-环，则打印 -1。

什么是独立集？在图 $G$ 中，独立集 $V'$ 是 $G$ 中一些顶点的集合，其中任意两个顶点之间都没有边相连。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

We have a simple undirected graph $G$ with $(S+T)$ vertices and $M$ edges. The vertices are numbered $1$ through $(S+T)$, and the edges are numbered $1$ through $M$. Edge $i$ connects Vertices $u_i$ and $v_i$.
Here, vertex sets $V_1 = \lbrace 1, 2,\dots, S\rbrace$ and $V_2 = \lbrace S+1, S+2, \dots, S+T \rbrace$ are both independent sets.

A cycle of length $4$ is called a 4-cycle.
If $G$ contains a 4-cycle, choose any of them and print the vertices in the cycle. You may print the vertices in any order.
If $G$ does not contain a 4-cycle, print -1.

What is an independent set? An independent set of a graph $G$ is a set $V'$ of some of the vertices in $G$ such that no two vertices of $V'$ have an edge between them.

Constraints

• $2 \leq S \leq 3 \times 10^5$
• $2 \leq T \leq 3000$
• $4 \leq M \leq \min(S \times T,3 \times 10^5)$
• $1 \leq u_i \leq S$
• $S + 1 \leq v_i \leq S + T$
• If $i \neq j$, then $(u_i, v_i) \neq (u_j, v_j)$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$S$ $T$ $M$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_M$ $v_M$

Output

If $G$ contains a 4-cycle, choose any of them, and print the indices of the four distinct vertices in the cycle. (The order of the vertices does not matter.)
If $G$ does not contain a 4-cycle, print -1.

Sample Input 1

2 3 5
1 3
1 4
1 5
2 4
2 5

Sample Output 1

1 2 4 5

There are edges between Vertices $1$ and $4$, $4$ and $2$, $2$ and $5$, and $5$ and $1$, so Vertices $1$, $2$, $4$, and $5$ form a 4-cycle. Thus, $1$, $2$, $4$, and $5$ should be printed.
The vertices may be printed in any order. Besides the Sample Output, 2 5 1 4 is also considered correct, for example.

Sample Input 2

3 2 4
1 4
1 5
2 5
3 5

Sample Output 2

-1

Some inputs may give $G$ without a 4-cycle.

Sample Input 3

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

Sample Output 3

1 7 2 9

update @ 2024/3/10 11:02:14