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 . γ c s ( 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 γ c s ( J ( G ) ) are related for the jump graph of various graph classes. The study further determines the exact value for γ c s ( J ( G ) ) of specific standard graphs. Additionally, the study characterizes γ c s ( J ( G ) ) = 2 and a tight bond is identified for γ c s ( J ( G ) ) , particularly for G with specific conditions.