Immediate versus eventual conversion: comparing geodetic and hull numbers in P3-convexity

Authors:
Carmen Cecilia Centeno;Lucia Draque Penso;Dieter Rautenbach;Vinícius Gusmcão Pereira de Sá
Affiliations:
Instituto de Matemática, NCE, and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil;Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany;Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany;Instituto de Matemática, NCE, and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil
Venue:
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
Year:
2012

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Abstract

We study the graphs G for which the hull number h(G) and the geodetic number g(G) with respect to P3-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.