Paired Disjunctive Domination Number of Middle Graphs
Hande Tuncel Golpek, Zeliha Kartal Yildiz, Aysun Aytac
TL;DR
The paper analyzes the paired disjunctive domination number of middle graphs $M(G)$, a parameter capturing redundancy in domination on a graph–edge hybrid. It derives exact values and tight bounds for $\\gamma_{pr}^{d}(M(G))$ across key graph families (paths, cycles, friendship, double stars, joins) and establishes structural results, such as reductions to subdivision vertices and deletion-based bounds, with a strong emphasis on trees. By combining combinatorial constructions with properties of $M(G)$, the authors illuminate how middle-graph transformations influence domination robustness. These results advance understanding of refined domination parameters on transformed graphs and have potential implications for designing fault-tolerant networks and related combinatorial problems.
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.