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G - Unique Walk

Score : $600$ points

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).

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.

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

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.

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

No

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