Score : $500$ points

问题陈述

你得到一个包含 $N$ 个顶点和 $M$ 条边的简单无向图。顶点编号从 $1$ 到 $N$，边编号从 $1$ 到 $M$。第 $i$ 条边连接顶点 $U_i$ 和顶点 $V_i$。

给定整数 $K, S, T$ 和 $X$，满足以下条件的序列 $A = (A_0, A_1, \dots, A_K)$ 有多少种？

• $A_i$ 是一个介于 $1$ 和 $N$（包括两端点）之间的整数。
• $A_0 = S$
• $A_K = T$
• 存在一条直接连接顶点 $A_i$ 和顶点 $A_{i+1}$ 的边。
• 整数 $X\ (X≠S,X≠T)$ 在序列 $A$ 中出现偶数次（可能为零）。

由于答案可能会非常大，求解答案对 $998244353$ 取模的结果。

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 from $1$ through $N$, and the edges are numbered from $1$ through $M$. Edge $i$ connects Vertex $U_i$ and Vertex $V_i$.

You are given integers $K, S, T$, and $X$. How many sequences $A = (A_0, A_1, \dots, A_K)$ are there satisfying the following conditions?

• $A_i$ is an integer between $1$ and $N$ (inclusive).
• $A_0 = S$
• $A_K = T$
• There is an edge that directly connects Vertex $A_i$ and Vertex $A_{i+1}$.
• Integer $X\ (X≠S,X≠T)$ appears even number of times (possibly zero) in sequence $A$.

Since the answer can be very large, find the answer modulo $998244353$.

Constraints

• All values in input are integers.
• $2≤N≤2000$
• $1≤M≤2000$
• $1≤K≤2000$
• $1≤S,T,X≤N$
• $X≠S$
• $X≠T$
• $1≤U_i<V_i≤N$
• If $i ≠ j$, then $(U_i, V_i) ≠ (U_j, V_j)$.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$ $S$ $T$ $X$

$U_1$ $V_1$

$U_2$ $V_2$

$\vdots$

$U_M$ $V_M$

Output

Print the answer modulo $998244353$.

Sample Input 1

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

Sample Output 1

4

The following $4$ sequences satisfy the conditions:

• $(1, 2, 1, 2, 3)$
• $(1, 2, 3, 2, 3)$
• $(1, 4, 1, 4, 3)$
• $(1, 4, 3, 4, 3)$

On the other hand, $(1, 2, 3, 4, 3)$ and $(1, 4, 1, 2, 3)$ do not, since there are odd number of occurrences of $2$.

Sample Input 2

6 5 10 1 2 3
2 3
2 4
4 6
3 6
1 5

Sample Output 2

0

The graph is not necessarily connected.

Sample Input 3

10 15 20 4 4 6
2 6
2 7
5 7
4 5
2 4
3 7
1 7
1 4
2 9
5 10
1 3
7 8
7 9
1 6
1 2

Sample Output 3

952504739

Find the answer modulo $998244353$.

update @ 2024/3/10 10:29:36