Score : $400$ points

问题描述

给定一个包含 $N$ 个顶点和 $M$ 条边的简单无向图 $G$（简单图不包含自环或多重边）。对于 $i = 1, 2, \ldots, M$，第 $i$ 条边连接顶点 $u_i$ 和顶点 $v_i$。

请输出满足以下两个条件的整数对 $(u, v)$ 的数量，其中 $1 \leq u < v \leq N$。

• 图 $G$ 中不存在连接顶点 $u$ 和顶点 $v$ 的边。
• 在图 $G$ 中添加一条连接顶点 $u$ 和顶点 $v$ 的边后，得到的图是一个二分图。

什么是二分图？

二分图 是一种无向图，其定义为：如果可以将每个顶点涂成黑色或白色，以满足以下条件，则称该图为二分图。

• 没有边连接颜色相同的顶点。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a simple undirected graph $G$ with $N$ vertices and $M$ edges (a simple graph does not contain self-loops or multi-edges). For $i = 1, 2, \ldots, M$, the $i$-th edge connects vertex $u_i$ and vertex $v_i$.

Print the number of pairs of integers $(u, v)$ that satisfy $1 \leq u \lt v \leq N$ and both of the following conditions.

• The graph $G$ does not have an edge connecting vertex $u$ and vertex $v$.
• Adding an edge connecting vertex $u$ and vertex $v$ in the graph $G$ results in a bipartite graph.

What is a bipartite graph?

An undirected graph is said to be bipartite if and only if one can paint each vertex black or white to satisfy the following condition.

• No edge connects vertices painted in the same color.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq \min \lbrace 2 \times 10^5, N(N-1)/2 \rbrace$
• $1 \leq u_i, v_i \leq N$
• The graph $G$ is simple.
• All values in the input are integers.

Input

The 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 the answer.

Sample Input 1

5 4
4 2
3 1
5 2
3 2

Sample Output 1

2

We have two pairs of integers $(u, v)$ that satisfy the conditions in the problem statement: $(1, 4)$ and $(1, 5)$. Thus, you should print $2$.
The other pairs do not satisfy the conditions. For instance, for $(1, 3)$, the graph $G$ has an edge connecting vertex $1$ and vertex $3$; for $(4, 5)$, adding an edge connecting vertex $4$ and vertex $5$ in the graph $G$ does not result in a bipartite graph.

Sample Input 2

4 3
3 1
3 2
1 2

Sample Output 2

0

Note that the given graph may not be bipartite or connected.

Sample Input 3

9 11
4 9
9 1
8 2
8 3
9 2
8 4
6 7
4 6
7 5
4 5
7 8

Sample Output 3

9

update @ 2024/3/10 11:50:13