REGNANT AND CAPTIVE DOMINATION IN SOME GENERALIZED GRAPHS

Abstract

The term "Regnant Domination" is presented in this study as a novel idea in graph dominance. Let $D^{\prime} \subseteq V(G)$. Then $D^{\prime}$ is an $\mathrm{RD}$-set if all vertices of $D^{\prime}$ are adjacent to at least a single vertex of $V-D^{\prime}$ except one, i.e., in the subset $D^{\prime}$, if exactly one vertex is not adjacent to any of the vertex of $V-D^{\prime}$, then $D^{\prime}$ is called an RD-set. We apply this new notion on helm graph to find regnant domination number for this graph. Further, we obtain some results related to captive domination defined by Al-Harere et al. [1] for some generalized graphs like Jahangir graph, helm graph, windmill graph and Petersen graph.

Received: July 19, 2022
Accepted: September 26, 2022