Heredity for generalized power domination

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 $γ\_{p,k}(G-e)$, $γ\_{p,k}(G/e)$ and for $γ\_{p,k}(G-v)$ in terms of $γ\_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $γ\_{p,k}(G-e) = γ\_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.

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