Edge Domination in Shadow Distance Graphs

Abstract

Let $G$ be a simple connected and undirected graph. The shadow graph of $G$, denoted $ D _{2}(G) $ is the graph constructed from $G$ by taking two copies of $G$ namely $G$ itself and $G^ {'}$ and by joining each vertex $u$ in $G$ to the neighbors of the corresponding vertex $u^{'}$ in $G^ {'}$. Let $D$ be the set of all distances between distinct pairs of vertices in $G$ and let $D_s$ $($called the distance set$)$ be a subset of $D$. The distance graph of $G$ denoted by $D(G,D_s)$ is the graph having the same vertex set as that of $G$ and two vertices $u$ and $v$ are adjacent in $D(G,D_s)$ whenever $d(u,v)\in D_s$. In this paper, we define a new graph called the shadow distance graph and determine the edge domination number of the shadow distance graph of the path graph, the cycle graph and the sunlet graph with specified distance sets.