Edge domination in various snake graphs

A set $F subseteq E(G) $ is an edge dominating set if each edge in $E(G)$ is either in $F$ or is adjacent to an edge in $F$. An edge dominating set $F$ is called a minimal edge dominating set if no proper subset $F ^ prime$ of $F$ is an edge dominating set. The edge domination number $gamma ^prime(G)$ is the minimum cardinality among all minimal edge dominating sets. We investigate the edge domination number of some graphs called snakes which are obtained from path $P_n$ by replacing its edges by cycles $C_3$ and $C_4$.