Generalized power domination in WK-Pyramid Networks

The notion of power domination arises in the context of monitoring an electric power system with as few phase measurement units as possible. The $k-$power domination number of a graph $G$ is the minimum cardinality of a $k-$power dominating set ($k-$PDS) of $G$. In this paper, we determine the $k-$power domination number of WK-Pyramid networks, $WKP_{(C,L)}$, for all positive values of $k$ except for $k=C-1, C \geq 2$, for which we give an upper bound. The $k-$propagation radius of a graph $G$ is the minimum number of propagation steps needed to monitor the graph $G$ over all minimum $k-$PDS. We obtain the $k-$propagation radius of $WKP_{(C,L)}$ in some cases.