Complement Spanning Trees

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### Read problem statements in [Hindi](http://www.codechef.com/download/translated/AUG19/hindi/CSTREE.pdf), [Bengali](http://www.codechef.com/download/translated/AUG19/bengali/CSTREE.pdf), [Mandarin Chinese](http://www.codechef.com/download/translated/AUG19/mandarin/CSTREE.pdf), [Russian](http://www.codechef.com/download/translated/AUG19/russian/CSTREE.pdf), and [Vietnamese](http://www.codechef.com/download/translated/AUG19/vietnamese/CSTREE.pdf) as well. You are given a small graph $G_s$ with $N$ nodes (numbered $1$ through $N$) and $M$ edges. Let's create a big graph $G_b$ with $NK$ nodes (numbered $1$ through $NK$) and $MK$ edges. $G_b$ consists of $K$ copies of $G_s$ ― formally, for each $k$ ($0 \le k \le K1$) and each edge between nodes $u$ and $v$ in $G_s$, there is an edge between nodes $u+kN$ and $v+kN$ in $G_b$. Afterwards, let's take the complement of $G_b$ and call it $H$ ― formally, $H$ is a graph with $NK$ nodes (numbered $1$ through $NK$) such that for each pair of nodes, there is an edge between these two nodes in $H$ if and only if there is no edge between these two nodes in $G_b$. Count the number of spanning trees in $H$. Since this number could be very large, compute it modulo $998,244,353$. ### Input  The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.  The first line of each test case contains three spaceseparated integers $N$, $M$ and $K$.  Each of the next $M$ lines contains two spaceseparated integers $a$ and $b$ denoting that the nodes $a$ and $b$ in $G_s$ are connected by an edge. ### Output For each test case, print a single line containing one integer ― the number of spanning trees in $H$ modulo $998,244,353$. ### Constraints  $1 \le T \le 20$  $1 \le N \le 30$  $0 \le M \le N(N1)/2$  $1 \le K \le 10^8$  $1 \le a, b \le N$  $G_s$ does not contain any selfloops or multiedges ### Subtasks **Subtask #1 (15 points):** $K \le 10$ **Subtask #2 (19 points):**  $K \le 50$  $M \le 5$ **Subtask #3 (21 points):**  $K \le 5,000$  $M = N(N1)/2$ **Subtask #4 (22 points):** $K \le 5,000$ **Subtask #5 (23 points):** original constraints ### Example Input ``` 5 3 0 1 3 3 2 1 2 1 3 2 3 3 3 1 1 2 1 3 2 3 6 9 20 1 3 2 3 4 3 5 3 6 3 1 4 2 4 4 5 4 6 30 4 5000 1 2 3 4 5 6 6 7 ``` ### Example Output ``` 3 81 0 13872131 797280964 ``` ### Explanation **Example case 1:** $H$ is the complete graph on three nodes, so the number of spanning trees is $3$ (any two of the edges form a spanning tree).Author:  lg5293 
Tags  lg5293 
Date Added:  24062019 
Time Limit:  3 sec 
Source Limit:  50000 Bytes 
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