102823: [AtCoder]ABC282 D - Make Bipartite 2

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.

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