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F - Well-defined Path Queries on a Namori

Score : $500$ points

Problem Statement

You are given a connected simple undirected graph $G$ with $N$ vertices numbered $1$ to $N$ and $N$ edges. The $i$-th edge connects Vertex $u_i$ and Vertex $v_i$ bidirectionally.

Answer the following $Q$ queries.

• Determine whether there is a unique simple path from Vertex $x_i$ to Vertex $y_i$ (a simple path is a path without repetition of vertices).

Constraints

• $3 \leq N \leq 2 \times 10^5$
• $1 \leq u_i < v_i\leq N$
• $(u_i,v_i) \neq (u_j,v_j)$ if $i \neq j$.
• $G$ is a connected simple undirected graph with $N$ vertices and $N$ edges.
• $1 \leq Q \leq 2 \times 10^5$
• $1 \leq x_i < y_i\leq N$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_N$ $v_N$

$Q$

$x_1$ $y_1$

$x_2$ $y_2$

$\vdots$

$x_Q$ $y_Q$

Output

Print $Q$ lines.

The $i$-th line $(1 \leq i \leq Q)$ should contain Yes if there is a unique simple path from Vertex $x_i$ to Vertex $y_i$, and No otherwise.

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

No
Yes
No

The simple paths from Vertex $1$ to $2$ are $(1,2)$ and $(1,3,2)$, which are not unique, so the answer to the first query is No.

The simple path from Vertex $1$ to $4$ is $(1,4)$, which is unique, so the answer to the second query is Yes.

The simple paths from Vertex $1$ to $5$ are $(1,2,5)$ and $(1,3,2,5)$, which are not unique, so the answer to the third query is No.

10
3 5
5 7
4 8
2 9
1 2
7 9
1 6
4 10
2 5
2 10
10
1 8
6 9
8 10
6 8
3 10
3 9
1 10
5 8
1 10
7 8

Yes
No
Yes
Yes
No
No
Yes
No
Yes
No