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.
Highlights
Domination is a well studied graph parameter, and a classical topic in graph theory
To address the problem of monitoring electrical networks with phasor measurement units (see Baldwin et al (1993)), power domination was introduced as a variation of the classical domination (see Haynes et al (2002))
The kpropagation radius of a graph G was introduced in Dorbec and Klavzar (2014) as a way to measure the efficiency of a minimum k-power dominating set (k-PDS)
Summary
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 γP,k(G − e), γP,k(G/e) and for γP,k(G − v) in terms of γP,k(G), and give examples for which these bounds are tight. We characterize all graphs for which γP,k(G − e) = γP,k(G) + 1 for any edge e. We consider the behaviour of the propagation radius of graphs by similar modifications
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
