102624: [AtCoder]ABC262 E - Red and Blue Graph

Problem Statement

You are given a simple undirected graph with $N$ vertices and $M$ edges. The vertices are numbered $1, \dots, N$, and the $i$-th $(1 \leq i \leq M)$ edge connects Vertices $U_i$ and $V_i$.

There are $2^N$ ways to paint each vertex red or blue. Find the number, modulo $998244353$, of such ways that satisfy all of the following conditions:

• There are exactly $K$ vertices painted red.
• There is an even number of edges connecting vertices painted in different colors.

Constraints

• $2 \leq N \leq 2 \times 10^5$
• $1 \leq M \leq 2 \times 10^5$
• $0 \leq K \leq N$
• $1 \leq U_i \lt V_i \leq N \, (1 \leq i \leq M)$
• $(U_i, V_i) \neq (U_j, V_j) \, (i \neq j)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$
$U_1$ $V_1$
$\vdots$
$U_M$ $V_M$

Output

Print the answer.

Sample Input 1

4 4 2
1 2
1 3
2 3
3 4

Sample Output 1

2

The following two ways satisfy the conditions.

• Paint Vertices $1$ and $2$ red and Vertices $3$ and $4$ blue.
• Paint Vertices $3$ and $4$ red and Vertices $1$ and $2$ blue.

In either of the ways above, the $2$-nd and $3$-rd edges connect vertices painted in different colors.

Sample Input 2

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

Sample Output 2

64