Edge domination in some brick product graphs

Abstract

Let G = (V, E) be a simple connected and undirected graph. A set F of edges in G is called an edge dominating set if every edge e in E − F is adjacent to at least one edge in F. The edge domination number γ′ (G) of G is the minimum cardinality of an edge dominating set of G. The shadow graph of G, denoted D₂(G) is the graph constructed from G by taking two copies of G, say G itself and G′ and joining each vertex u in G to the neighbors of the corresponding vertex u′ in G′. Let D be the set of all distinct pairs of vertices in G and let Ds (called the distance set) be a subset of D. The distance graph of G, denoted by D(G, Ds) is the graph having the same vertex set as that of G and two vertices u and v are adjacent in D(G, Ds) whenever d(u, v) ∈ Ds. In this paper, we determine the edge domination number of the shadow distance graph of the brick product graph C(2n, m, r).