Let $G=(V,E)$ be a graph having all the following restrictions:
- Every vertex $v \in V$ has degree $4$.
- Every vertex $v \in V$ belongs to at least $2$ triangles.
- For every vertex $v \in V$, if $v$ belongs to exactly $2$ triangles then such $2$ triangles do not share any edge.
I would like to know whether such kind of graphs have been studied, what is known about them, and in particular which is the computational complexity of counting vertex covers and edge covers (even modulo $2$) on them.