A study of local domination number of Sn H graph

Abstract

All graphs in this paper are undirected, connected and simple graph. Let G = (V,E) be a graph of order |V| and size |E|. We define a set D as a dominating set if for every vertex μ epsilon V – D is adjacent to some vertex ν epsilon D. The domination number γ(G) is the minimum cardinality of dominating set. By a locating dominating set of graph G = (V, E), we define for every two vertices μ,ν epsilon V(G) – D, N(ν) bigcap D ≠ Ø. Locating dominating set is a special case of dominating set with an extra constrain above. The minimum cardinality of a locating dominating set is locating dominating number γ L (G). The value of locating dominating number is γ L (G) ⊆ V (G). This paper studies locating dominating set of edge comb product of graphs, denoted by GH. The graph G rtrie H is a graph obtained by taking one copy of G and |E(G)| copies of H and grafting the i-th copy of H at the edge e to the i-th edge of G, where G is star graph S n and H is any special graph.