The Geodetic Number of Cm × Cn

Abstract

Graph theory is the theoretical fundamental of network technologies. This paper reports an investigation of stability issues of the network technologies based on the graph theory. It also presents solutions of the geodetic number of graph Cm × Cn. There are many parameters which can measure to some extent the robustness of an interconnection network with respect to link failures, such as domination number and bondage number. Another parameter, the geodetic number, has been studied widely in the recent literature. Determining the geodetic number of an arbitrary graph however can be proved to be NP-hard. Therefore it is natural to investigate some specific cases for the geodetic number of special graphs. The solving process of one of such cases is presented. Index Terms—geodesic, geodetic set, geodetic number, con- nected geodetic set, connected geodetic number, open geodetic set, open geodetic number.