The lower and upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori

Abstract

A vertex set D in a graph G is called a geodetic set if each vertex of G is lying on some shortest u−v path of G, where u, v∈D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this article, we find out the geodetic numbers, the lower and the upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori.

Keywords:

• geodetic number
• forcing geodetic number
• complete n-partite graphs
• meshes
• tori

2000 AMS Subject Classifications :

• 05C25
• 68Q25
• 68W40
• 68R10

Acknowledgements

The author wishes to thank the anonymous referees for their comments and suggestions. This research was partially supported by National Science Council under the Grant NSC97-2221-E-034-012.