Score : $600$ points

问题描述

你得到一个简单连通的无向图 $G$，包含 $N$ 个顶点和 $M$ 条边。
$G$ 中的顶点编号为顶点 $1$、顶点 $2$、$\ldots$ 和顶点 $N$，其边编号为边 $1$、边 $2$、$\ldots$ 和边 $M$。其中边 $i$ 连接顶点 $U_i$ 和顶点 $V_i$。
同时，你还得到了图中边的一个子集：$S=\{x_1,x_2,\ldots,x_K\}$。

判断在图 $G$ 上是否存在一条路径，使得对于所有 $x \in S$，路径恰好包含边 $x$ 一次。
该路径可以任意次数（包括零次）包含不属于集合 $S$ 的边。

什么是路径？在一个无向图 $G$ 上，路径是一个由 $k$ 个顶点（$k$ 是正整数）和 $(k-1)$ 条边交替组成的序列，即 $v_1,e_1,v_2,\ldots,v_{k-1},e_{k-1},v_k$，其中边 $e_i$ 连接顶点 $v_i$ 和顶点 $v_{i+1}$。序列中可以包含相同边或顶点多次。若存在且仅存在一个 $1\leq i\leq k-1$ 满足 $e_i=x$，则称这条路径恰好包含边 $x$ 一次。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a simple connected undirected graph $G$ with $N$ vertices and $M$ edges.
The vertices of $G$ are numbered vertex $1$, vertex $2$, $\ldots$, and vertex $N$, and its edges are numbered edge $1$, edge $2$, $\ldots$, and edge $M$. Edge $i$ connects vertex $U_i$ and vertex $V_i$.
You are also given a subset of the edges: $S=\{x_1,x_2,\ldots,x_K\}$.

Determine if there is a walk on $G$ that contains edge $x$ exactly once for all $x \in S$.
The walk may contain an edge not in $S$ any number of times (possibly zero).

What is a walk? A walk on an undirected graph $G$ is a sequence consisting of $k$ vertices ($k$ is a positive integer) and $(k-1)$ edges occurring alternately, $v_1,e_1,v_2,\ldots,v_{k-1},e_{k-1},v_k$, such that edge $e_i$ connects vertex $v_i$ and vertex $v_{i+1}$. The sequence may contain the same edge or vertex multiple times. A walk is said to contain an edge $x$ exactly once if and only if there is exactly one $1\leq i\leq k-1$ such that $e_i=x$.

Constraints

• $2 \leq N \leq 2\times 10^5$
• $N-1 \leq M \leq \min(\frac{N(N-1)}{2},2\times 10^5)$
• $1 \leq U_i<V_i\leq N$
• If $i\neq j$, then $(U_i,V_i)\neq (U_j,V_j)$ .
• $G$ is connected.
• $1 \leq K \leq M$
• $1 \leq x_1<x_2<\cdots<x_K \leq M$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$U_1$ $V_1$

$U_2$ $V_2$

$\vdots$

$U_M$ $V_M$

$K$

$x_1$ $x_2$ $\ldots$ $x_K$

Output

Print Yes if there is a walk satisfying the condition in the Problem Statement; print No otherwise.

Sample Input 1

6 6
1 3
2 3
3 4
4 5
4 6
5 6
4
1 2 4 5

Sample Output 1

Yes

The walk $(v_1,e_1,v_3,e_3,v_4,e_4,v_5,e_6,v_6,e_5,v_4,e_3,v_3,e_2,v_2)$ satisfies the condition, where $v_i$ denotes vertex $i$ and $e_i$ denotes edge $i$.
In other words, the walk travels the vertices on $G$ in this order: $1\to 3\to 4\to 5\to 6\to 4\to 3\to 2$.
This walk satisfies the condition because it contains edges $1$, $2$, $4$, and $5$ exactly once each.

Sample Input 2

6 5
1 2
1 3
1 4
1 5
1 6
3
1 2 3

Sample Output 2

No

There is no walk that contains edges $1$, $2$, and $3$ exactly once each, so No should be printed.

update @ 2024/3/10 11:58:49