An efficient algorithm to find a shortest cycle including a specific vertex
You can find the shortest cycle that contains $v$ by running BFS from $v$, and stopping at the first time that $v$ is reached again (or if BFS terminated without reaching $v$).

An important property of BFS is that if it runs from vertex $v$, then every vertex $s$, when it is first reached, then the path that was found from $v$ to $s$ is minimal. Thus, reaching $v$ from $v$ with BFS finds the shortest path from $v$ to itself, namely the shortest cycle that contains $v$.