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Some remarks on the geodetic number of a graph |
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| Authors: | Mitre C Dourado |
| Institution: | a ICE, Universidade Federal Rural do Rio de Janeiro and NCE - UFRJ, Brazil b Universidade Federal Fluminense, Instituto de Computação, Niterói, RJ, Brazil c Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany d Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brazil |
| Abstract: | 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 |
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