$(1)$ Let $G$ be a graph on vertices $v_1,$dots,$v_{12}$ with the following adjacency matrix:

$([0,1,1,1,1,0,0,0,0,0,0,0],[1,0,1,1,1,0,0,0,0,0,0,0],[1,1,0,1,1,0,0,0,0,0,0,0],[1,1,1,0,1,1,0,0,0,0,0,0],[1,1,1,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,1,1,0,0],[0,0,0,0,0,1,1,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,1,0,0],[0,0,0,0,0,0,1,1,1,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,0,1,0])$

Use the connectedness algorithm to determine the connected component of $v_1.$ State whether or not $G$ is connected and write down all its connected components.

We can extend the notion of a bridge from the lecture notes to mean an edge whose removal causes the number of connected components to increase. Does $G$ contain any bridges in this sense? List them all if so$.$