The relationship between $k$-forcing and $k$-power domination

TL;DR

This paper explores the relationship between $k$-forcing and $k$-power domination in graphs, providing bounds and methods for their computation, thus unifying and extending previous concepts in graph theory.

Contribution

It establishes a new relationship between $k$-forcing and $k$-power domination numbers, and introduces contraction techniques for their parallel computation.

Findings

Bounds one parameter in terms of the other.

Provides contraction-based methods for computing $k$-forcing and $k$-power dominating sets.

Unifies the study of $k$-forcing and $k$-power domination.

Abstract

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. Chang et al. introduced $k$-power domination in [Generalized power domination in graphs, {\it Discrete Applied Math.} 160 (2012) 1691-1698] as a generalization of power domination and standard graph domination. Independently, Amos et al. defined $k$-forcing in [Upper bounds on the $k$-forcing number of a graph, {\it Discrete Applied Math.} 181 (2015) 1-10] to generalize zero forcing. In this paper, we combine the study of $k$-forcing and $k$-power domination, providing a new approach to analyze both processes. We give a relationship between the $k$-forcing and the $k$-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of $k$-forcing and $k$-power dominating sets.