Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs. Keywords. Cograph; convex hull; convex set; geodetic number; split graph; unit interval graph

• Type: Journal
• Year: 2010
• Where: DM
• Authors: Mitre Costa Dourado, Fábio Protti, Dieter Rautenbach, Jayme Luiz Szwarcfiter
• Keywords: DM 2010, Geodetic, Geodetic Number, Geodetic Set