CF1761E.Make It Connected

You are given a simple undirected graph consisting of $n$ vertices. The graph doesn't contain self-loops, there is at most one edge between each pair of vertices. Your task is simple: make the graph connected.

You can do the following operation any number of times (possibly zero):

• Choose a vertex $u$ arbitrarily.
• For each vertex $v$ satisfying $v\ne u$ in the graph individually, if $v$ is adjacent to $u$ , remove the edge between $u$ and $v$ , otherwise add an edge between $u$ and $v$ .

Find the minimum number of operations required to make the graph connected. Also, find any sequence of operations with the minimum length that makes the graph connected.

Each test contains multiple test cases. The first line contains a single integer $t$ ( $1\leq t\leq 800$ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains a single integer $n$ ( $2\leq n\leq 4000$ ) — the number of vertices in the graph.

Then $n$ lines follow. The $i$ -th row contains a binary string $s_i$ of length $n$ , where $s_{i,j}$ is '1' if there is an edge between vertex $i$ and $j$ initially, otherwise $s_{i,j}$ is '0'.

It is guaranteed that $s_{i,i}$ is always '0' and $s_{i,j}=s_{j,i}$ for $1\leq i,j\leq n$ .

It is guaranteed that the sum of $n$ over all test cases does not exceed $4000$ .

For each test case, in the first line, output an integer $m$ — the minimum number of operations required.

If $m$ is greater than zero, then print an extra line consisting of $m$ integers — the vertices chosen in the operations in your solution. If there are multiple solutions with the minimum number of operations, print any.

In the first test case, the graph is connected at the beginning, so the answer is $0$ .

In the second test case, if we do the operation with vertex $1$ , we will get the following graph represented by an adjacency matrix:

$$\begin{bmatrix} 0&1&1\\ 1&0&1\\ 1&1&0 \end{bmatrix}$$

It's obvious that the graph above is connected.

In the third test case, if we do the operation with vertex $3$ and $4$ , we will get the following graph represented by an adjacency matrix:

$$\begin{bmatrix} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{bmatrix}$$

It's obvious that the graph above is connected, and it can be proven that we can't perform less than $2$ operations to make the graph connected.