Connected power domination number of product graphs

TL;DR

This paper investigates the connected power domination number in various graph products, providing exact values and bounds for specific cases, advancing understanding of this parameter in graph theory.

Contribution

It derives exact values and tight bounds for the connected power domination number in Cartesian, tensor, and other graph products, extending previous results.

Findings

Exact value for $ ext{connected power domination}$ of $G owtie H$ for non-trivial graphs.

Tight upper bounds for the Cartesian product's connected power domination number.

Discussion of the connected power domination number in tensor product graphs.

Abstract

In this paper, we consider the connected power domination number ($\gamma_{P, c}$) of three standard graph products. The exact value for $\gamma_{P, c}(G\circ H)$ is obtained for any two non-trivial graphs $G$ and $H.$ Further, tight upper bounds are proved for the connected power domination number of the Cartesian product of two graphs $G$ and $H.$ Consequently, the exact value of the connected power domination number of the Cartesian product of some standard graphs is determined. Finally, the connected power domination number of tensor product of graphs is discussed.