Paired Disjunctive Domination Number of Middle Graphs

TL;DR

This paper investigates the paired disjunctive domination number in middle graphs, providing bounds, exact values for specific classes, and insights into its behavior in various graph transformations.

Contribution

It introduces new bounds and exact values for the paired disjunctive domination number in middle graphs across multiple graph classes and operations.

Findings

Bounds established for arbitrary graphs and trees.

Exact values computed for specific graph classes.

Results on middle graphs from join operations.

Abstract

The concept of domination in graphs plays a central role in understanding structural properties and applications in network theory. In this study, we focus on the paired disjunctive domination number in the context of middle graphs, a transformation that captures both adjacency and incidence relations of the original graph. We begin by investigating this parameter for middle graphs of several special graph classes, including path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs, star graphs, friendship graphs, and double star graphs. We then present general results by establishing lower and upper bounds for the paired disjunctive domination number in middle graphs of arbitrary graphs, with particular emphasis on trees. Additionally, we determine the exact value of the parameter for middle graphs obtained through the join operation. These findings contribute to the broader understanding of domination-type parameters in transformed graph structures and offer new insights into their combinatorial behavior.