Counting Shortcuts
Description
Given an undirected connected graph with $n$ vertices and $m$ edges. The graph contains no loops (edges from a vertex to itself) and multiple edges (i.e. no more than one edge between each pair of vertices). The vertices of the graph are numbered from $1$ to $n$.
Find the number of paths from a vertex $s$ to $t$ whose length differs from the shortest path from $s$ to $t$ by no more than $1$. It is necessary to consider all suitable paths, even if they pass through the same vertex or edge more than once (i.e. they are not simple).
Graph consisting of $6$ of vertices and $8$ of edges For example, let $n = 6$, $m = 8$, $s = 6$ and $t = 1$, and let the graph look like the figure above. Then the length of the shortest path from $s$ to $t$ is $1$. Consider all paths whose length is at most $1 + 1 = 2$.
- $6 \rightarrow 1$. The length of the path is $1$.
- $6 \rightarrow 4 \rightarrow 1$. Path length is $2$.
- $6 \rightarrow 2 \rightarrow 1$. Path length is $2$.
- $6 \rightarrow 5 \rightarrow 1$. Path length is $2$.
There is a total of $4$ of matching paths.
The first line of test contains the number $t$ ($1 \le t \le 10^4$) —the number of test cases in the test.
Before each test case, there is a blank line.
The first line of test case contains two numbers $n, m$ ($2 \le n \le 2 \cdot 10^5$, $1 \le m \le 2 \cdot 10^5$) —the number of vertices and edges in the graph.
The second line contains two numbers $s$ and $t$ ($1 \le s, t \le n$, $s \neq t$) —the numbers of the start and end vertices of the path.
The following $m$ lines contain descriptions of edges: the $i$th line contains two integers $u_i$, $v_i$ ($1 \le u_i,v_i \le n$) — the numbers of vertices that connect the $i$th edge. It is guaranteed that the graph is connected and does not contain loops and multiple edges.
It is guaranteed that the sum of values $n$ on all test cases of input data does not exceed $2 \cdot 10^5$. Similarly, it is guaranteed that the sum of values $m$ on all test cases of input data does not exceed $2 \cdot 10^5$.
For each test case, output a single number — the number of paths from $s$ to $t$ such that their length differs from the length of the shortest path by no more than $1$.
Since this number may be too large, output it modulo $10^9 + 7$.
Input
The first line of test contains the number $t$ ($1 \le t \le 10^4$) —the number of test cases in the test.
Before each test case, there is a blank line.
The first line of test case contains two numbers $n, m$ ($2 \le n \le 2 \cdot 10^5$, $1 \le m \le 2 \cdot 10^5$) —the number of vertices and edges in the graph.
The second line contains two numbers $s$ and $t$ ($1 \le s, t \le n$, $s \neq t$) —the numbers of the start and end vertices of the path.
The following $m$ lines contain descriptions of edges: the $i$th line contains two integers $u_i$, $v_i$ ($1 \le u_i,v_i \le n$) — the numbers of vertices that connect the $i$th edge. It is guaranteed that the graph is connected and does not contain loops and multiple edges.
It is guaranteed that the sum of values $n$ on all test cases of input data does not exceed $2 \cdot 10^5$. Similarly, it is guaranteed that the sum of values $m$ on all test cases of input data does not exceed $2 \cdot 10^5$.
Output
For each test case, output a single number — the number of paths from $s$ to $t$ such that their length differs from the length of the shortest path by no more than $1$.
Since this number may be too large, output it modulo $10^9 + 7$.
Samples
4
4 4
1 4
1 2
3 4
2 3
2 4
6 8
6 1
1 4
1 6
1 5
1 2
5 6
4 6
6 3
2 6
5 6
1 3
3 5
5 4
3 1
4 2
2 1
1 4
8 18
5 1
2 1
3 1
4 2
5 2
6 5
7 3
8 4
6 4
8 7
1 4
4 7
1 6
6 7
3 8
8 5
4 5
4 3
8 2
2
4
1
11
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