Heredity for generalized power domination

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.

Discrete Mathematics and Theoretical Computer Science, 2016