Strong domination integrity in graphs and fuzzy graphs

Abstract

Let H = (V, E) be a graph and xy ∈ E (H). Then x strongly dominates y if deg(x) ⩾ deg(y). A subset S of V is said to be a strong dominating set if every node y ∈ V – S is strongly dominated by some node x in H and is denoted by sd-set. The strong domination number γs (H) is the minimum cardinality of a strong dominating set. In this paper, we introduce a new vulnerability parameter called strong domination integrity in graphs. Strong domination integrity of some families of graphs are determined and its bounds are also obtained. The proposed parameter is applied in water distribution network system to identify the influential group of nodes within the network. Fuzzy graphs can be used to model uncertain networks. By using membership values of strong arcs, strong domination integrity is extended to fuzzy graphs as a new vulnerability parameter. In this study, we investigate the strong domination integrity for complete bipartite fuzzy graphs, complete fuzzy graphs and bounds are also derived. Some basic results and theorems are obtained. This vulnerability parameter is also applied in the transportation network systems.