Geodetic Number and Geo-Chromatic Number of 2-Cartesian Product of Some Graphs

Abstract

A set S ⊆ V (G) is called a geodetic set if every vertex of G lies on a shortest u-v path for some u, v ∈ S , the minimum cardinality among all geodetic sets is called geodetic number and is denoted by . A set C ⊆ V (G) is called a chromatic set if C contains all vertices of different colors in G , the minimum cardinality among all chromatic sets is called the chromatic number and is denoted by . A geo-chromatic set S c ⊆ V (G ) is both a geodetic set and a chromatic set. The geo-chromatic number of G is the minimum cardinality among all geo-chromatic sets of G . In this paper, we determine the geodetic number and the geo-chromatic number of 2-cartesian product of some standard graphs like complete graphs, cycles and paths.