A1004. Minimizing Edges--Platinum

Bessie has a connected, undirected graph $G$ with $N$ vertices labeled
$1\ldots N$ and $M$ edges ($2\le N\le 10^5, N-1\le M\le \frac{N^2+N}{2}$). $G$
may contain self-loops (edges from nodes back to themselves), but no parallel
edges (multiple edges connecting the same endpoints).
Let $f_G(a,b)$ be a boolean function that evaluates to true if there exists a
path from vertex $1$ to vertex $a$ that traverses exactly $b$ edges for each
$1\le a\le N$ and $0\le b$, and false otherwise. If an edge is traversed
multiple times, it is included that many times in the count.
Elsie wants to copy Bessie. In particular, she wants to construct an
undirected graph $G'$ such that $f_{G'}(a,b)=f_G(a,b)$ for all $a$ and $b$.
Elsie wants to do the least possible amount of work, so she wants to construct
the smallest possible graph. Therefore, your job is to compute the minimum
possible number of edges in $G'$.
Each input contains $T$ ($1\le T\le 5\cdot 10^4$) test cases that should be
solved independently. It is guaranteed that the sum of $N$ over all test cases
does not exceed $10^5$, and the sum of $M$ over all test cases does not exceed
$2\cdot 10^5$.

The first line of the input contains $T$, the number of test cases.
The first line of each test case contains two integers $N$ and $M$.
The next $M$ lines of each test case each contain two integers $x$ and $y$
($1\le x\le y\le N$), denoting that there exists an edge between $x$ and $y$
in $G$.
Consecutive test cases are separated by newlines for readability.

For each test case, the minimum possible number of edges in $G'$ on a new
line.

In the first test case, Elsie can construct $G'$ by starting with $G$ and
removing $(2,5)$. Or she could construct a graph with the following edges,
since she isn't restricted to just removing edges from $G$:
1 2
1 4
4 3
4 5
Elsie definitely cannot do better than $N-1$ since $G'$ must also be
connected.