Co-Secure Domination in Jump Graphs for Enhanced Security

Abstract

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.