Immediate versus eventual conversion: Comparing geodetic and hull numbers in P 3-convexity

Abstract

We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P 3-convexity coincide. These two parameters correspond to the minimum cardinality of a set U of vertices of G such that the simple expansion process that iteratively adds to U, all vertices outside of U that have two neighbors in U, produces the whole vertex set of G either eventually or after one iteration, respectively. We establish numerous structural properties of the graphs G with h(G)∈=∈g(G), which allow the constructive characterization as well as the efficient recognition of all triangle-free such graphs. Furthermore, we characterize the graphs G that satisfy h(H)∈=∈g(H) for every induced subgraph H of G in terms of forbidden induced subgraphs. © 2012 Springer-Verlag Berlin Heidelberg.