The Domination number of 2-neighbourhood-corona graphs

Abstract

Given a simple graph G. Vertex set of G is V, and edge set of G is E. Domination set, denoted by S, that is subset of V such that every vertex in V which is not element of S has distance one to S. The least number of the elements of S is the domination number of the graph G, that is ϒ(G). Let G1 and G2 be a simple graph. G1 has n1 vertices, and has m1 edges. G2 has n2 vertices, and has m2 edges. We defined an operator called neighbourhood corona, denoted by a star ‘*’. Graph G1*G2 is a new graph obtained by making ni copies of second graph and for each i make connecting all vertices in i-th copy of second graph G2 to neighbours of vi = 1, 2, …, n. Furthermore, new graph 2-neighbourhood corona G1*2G2, has n1 copies of G2 and for each i make connecting to all vertices of ith copy of G2 to neighbours of vi, i = 1, 2, …, n. In this research, we determined ϒ(Gi*2G2) where G1 is a complete graph Kn or Cycle Cn, and G2 is K1 or P2. Furthermore, we determined ϒ(G1 *mG2) due to domination number of complete graph Kn and cycle Cn. Since ϒ(Kn) = 1 then ϒ (Kn*mK1) = 1 + m. Furthermore, ϒ(Kn*mP2) = 1 + m. Since then . Furthermore, .