Power Domination in Tree Derived Architectures

Abstract

A set $S$ of vertices in a graph $G$ is called a dominating set of $G$ if every vertex in $V(G)\mathrm{\S }$ is adjacent to some vertex in $S$. A set S is said to be a power dominating set of $G$ if every vertex in the system is monitored by the set $S$ following a set of rules for power system monitoring. A zero forcing set of $G$ is a subset of vertices B such that if the vertices in $B$ are colored blue and the remaining vertices are colored white initially, repeated application of the color change rule can color all vertices of $G$ blue. The power domination number and the zero forcing number of G are the minimum cardinality of a power dominating set and the minimum cardinality of a zero forcing set respectively of $G$. In this paper, we obtain the power domination number, total power domination number, zero forcing number and total forcing number for m-rooted sibling trees, l-sibling trees and I-binary trees. We also solve power domination number for circular ladder, Möbius ladder, and extended cycle-of-ladder.