Geodetic domination integrity in fuzzy graphs

Abstract

Let N = (V, E) be a simple graph and let X be a subset of V (N). If every node not in X lies on a geodesic path between two nodes from X then it is called a geodetic set. The geodetic number g (N) is the minimum cardinality of such set X. The subset X is called a dominating set if every node not in X has at least one neighbour in X. The minimum number of nodes of a dominating set is known as domination number γ (N). If the subset X is a geodetic set as well as a dominating set then it is called a geodetic dominating set. The minimum cardinality of a geodetic dominating set is known as geodetic domination number γg (N). The geodetic domination integrity of N is defined to be DIg (N) = min {|X| + m (N - X) : X is a geodetic dominating set of N}, where m (N - X) denotes the order of the largest component of N - X. Uncertain networks can be modelled using fuzzy graphs. In a graph, each vertex and each edge are equally significant. However, in fuzzy graphs, each vertex and each edge is important in terms of fuzziness in their own right. In this study, the concepts of geodetic dominating sets in fuzzy graphs and geodetic domination number are defined and bounds are obtained. Moreover, the vulnerability parameter Geodetic domination integrity is introduced in fuzzy graphs. Further, the geodetic domination integrity for complete fuzzy graphs, complete bipartite fuzzy graphs, Cartesian product of two strong fuzzy graphs and bounds are also discussed. The applications of this parameter are applied to a telecommunication network system model to identify the key persons in the system and applied in a fuzzy social network to find the most influential group within the network.