Tom and Jerry

As usual, Tom is trying to catch Jerry. To accomplish this task, Tom hired $K$ other cats (Tom does not take part in the chase). Formally, they are in a graph $G$ with $N$ vertices and $M$ edges. Let's denote the set of vertices of $G$ by $V$. The graph has a special property: for each simple cycle containing at least four vertices, there is an edge which is not part of this cycle, but connects tw