Heredity for generalized power domination

TL;DR

This paper investigates how the generalized power domination number of a graph changes under small modifications like edge and vertex deletion or contraction, providing bounds and characterizations.

Contribution

It establishes optimal bounds for the generalized power domination number after graph modifications and characterizes graphs with specific properties related to these bounds.

Findings

Bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$, and $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$

Characterization of graphs where $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$

Analysis of the propagation radius behavior under similar graph modifications

Abstract

In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma\_{p,k}(G-e)$, $\gamma\_{p,k}(G/e)$ and for $\gamma\_{p,k}(G-v)$ in terms of $\gamma\_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma\_{p,k}(G-e) = \gamma\_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.