Score : $300$ points

问题陈述

你得到一个含有 $N$ 个顶点和 $M$ 条边的简单无向图。顶点编号为 $1, 2, \dots, N$，边编号为 $1, 2, \dots, M$。 其中第 $i$ 条边 $(i = 1, 2, \dots, M)$ 连接顶点 $u_i$ 和 $v_i$。

确定该图是否是一条路径图。

简单无向图是什么？简单无向图是指不含自环且不存在多条边的图，其边没有方向。

路径图是什么？如果存在一个序列 $(v_1, v_2, \dots, v_N)$，它是 $(1, 2, \dots, N)$ 的一个排列，并满足以下条件，则具有 $N$ 个顶点（编号为 $1, 2, \dots, N$）的图被称作路径图：

• 对于所有 $i = 1, 2, \dots, N-1$，存在一条连接顶点 $v_i$ 和 $v_{i+1}$ 的边。
• 若整数 $i$ 和 $j$ 满足 $1 \leq i, j \leq N$ 以及 $|i - j| \geq 2$，则不存在连接顶点 $v_i$ 和 $v_j$ 的边。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1, 2, \dots, N$, and the edges are numbered $1, 2, \dots, M$.
Edge $i \, (i = 1, 2, \dots, M)$ connects vertices $u_i$ and $v_i$.

Determine if this graph is a path graph.

What is a simple undirected graph? A simple undirected graph is a graph without self-loops or multiple edges whose edges do not have a direction.

What is a path graph? A graph with $N$ vertices numbered $1, 2, \dots, N$ is said to be a path graph if and only if there is a sequence $(v_1, v_2, \dots, v_N)$ that is a permutation of $(1, 2, \dots, N)$ and satisfies the following conditions:

• For all $i = 1, 2, \dots, N-1$, there is an edge connecting vertices $v_i$ and $v_{i+1}$.
• If integers $i$ and $j$ satisfies $1 \leq i, j \leq N$ and $|i - j| \geq 2$, then there is no edge that connects vertices $v_i$ and $v_j$.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq 2 \times 10^5$
• $1 \leq u_i, v_i \leq N \, (i = 1, 2, \dots, M)$
• All values in the input are integers.
• The graph given in the input is simple.

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 Yes if the given graph is a path graph; print No otherwise.

Sample Input 1

4 3
1 3
4 2
3 2

Sample Output 1

Yes

Illustrated below is the given graph, which is a path graph.

Sample Input 2

2 0

Sample Output 2

No

Illustrated below is the given graph, which is not a path graph.

Sample Input 3

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

Sample Output 3

No

Illustrated below is the given graph, which is not a path graph.

update @ 2024/3/10 11:59:46