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 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 also consider the behaviour of the propagation radius of graphs by similar modifications.