Geodetic Domination in the Corona and Join of Graphs

Abstract

A subset S of vertices in a graph G is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A subset D of vertices in G is called dominating set if every vertex not in D has at least one neighbor in D. A geodetic dominating set S is both a geodetic and a dominating set. The geodetic (domination, geodetic domination) number g (G) (γ (G), γg (G)) of G is the minimum cardinality among all geodetic (dominating, geodetic dominating) sets in G. Let G and H be two graphs and let n be the order of G. The corona product G ∘ H is defined as the graph obtained from G and H by taking one copy of G and n copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H. In this paper we characterized the geodetic dominating sets in the corona and the join of two graphs. We show that if G is a connected graph of order n and H is a non-complete graph, then γg(G ∘ H) = nγg(K1 ∘ H).