Generalized power domination in WK-Pyramid Networks

TL;DR

This paper determines the $k$-power domination number and propagation radius of WK-Pyramid networks, providing exact values and bounds, which aids in efficient monitoring of electric power systems.

Contribution

It offers the first comprehensive analysis of $k$-power domination in WK-Pyramid networks, including exact numbers and bounds for various parameters.

Findings

Exact $k$-power domination numbers for most cases.

Upper bounds for $k$-power domination when $k=C-1$.

Some cases for $k$-propagation radius are explicitly determined.

Abstract

The notion of power domination arises in the context of monitoring an electric power system with as few phase measurement units as possible. The $k-$power domination number of a graph $G$ is the minimum cardinality of a $k-$power dominating set ($k-$PDS) of $G$. In this paper, we determine the $k-$power domination number of WK-Pyramid networks, $WKP_{(C,L)}$, for all positive values of $k$ except for $k=C-1, C \geq 2$, for which we give an upper bound. The $k-$propagation radius of a graph $G$ is the minimum number of propagation steps needed to monitor the graph $G$ over all minimum $k-$PDS. We obtain the $k-$propagation radius of $WKP_{(C,L)}$ in some cases.