Geodetic domination integrity in graphs

Abstract

Reciprocal version of product degree distance of cactus graphs Let G be a simple graph. A subset S ? V (G) is a said to be a geodetic set if every vertex u /? S lies on a shortest path between two vertices from S. The minimum cardinality of such a set S is the geodetic number g(G) of G. A subset D ? V (G) is a dominating set of G if every vertex u /? D has at least one neighbor in D. The domination number ?(G) is the minimum cardinality of a dominating set of G. A subset is said to be a geodetic dominating set of G if it is both a geodetic and a dominating set. The geodetic domination number ?g(G) is the minimum cardinality among all geodetic dominating sets in G. The geodetic domination integrity of a graph G is defined by DIg(G) = min{|S| + m(G ? S) : S is a geodetic dominating set of G}, where m(G ? S) denotes the order of the largest component in G?S. In this paper, we study the concepts of geodetic dominating integrity of some families of graphs and derive some bounds for the geodetic domination integrity. Also we obtain geodetic domination integrity of some cartesian product of graphs.