On the geodetic and the hull numbers in strong product graphs

Jose Caceres, Carmen Hernando, Merce Mora, Ignacio M. Pelayo, Maria Luz Puertas

Published 2009-12-16, updated 2010-06-07Version 3

A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the cardinality of a minimum geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also stablish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.