Co-Secure Domination in Jump Graphs for Enhanced Security

This study proposes a general approach to protect graphs using co-secure domination within jump graphs. In the context of graphs, a dominating set is a group of vertices that are either directly linked or connected to all other vertices within the graph. The minimum cardinality of the dominating set in a graph G is called the domination number γ (G). A set S⊆V of a graph G is called a co-secure dominating set, if, for all u∈S, there exists a node v∈N (u) and in V∖S so that (S∖u) ∪v dominates the graph G. γcs (G), the co-secure domination number, is the cardinality of a co-secure dominating set with minimum vertices within the graph G. It is a notable protective strategy in which the nodes that are attacked or damaged in an interconnection network can be replaced with alternative nodes to ensure network security. In a jump graph J (G), the vertices are the edges of G and the adjacency of the vertices of J (G) are given by the condition that these edges are not adjacent in G. This paper explains how γ (G) and γcs (J (G) ) are related for the jump graph of various graph classes. The study further determines the exact value for γcs (J (G) ) of specific standard graphs. Additionally, the study characterizes γcs (J (G) ) =2 and a tight bond is identified for γcs (J (G) ), particularly for G with specific conditions.