Convex dominating-geodetic partitions in graphs

Abstract

The distance d(u, v) between two vertices u and v in a connected graph G is the length of aThe convex domination number γ con (G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v S lies on a x − y geodesic between two vertices x, y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by (G). Let D, S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a γ con (G)-set and S is a (G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.