Bounds on the propagation radius in power domination

Let $G$ be a graph and let $S\subseteq V(G)$. It is said that $S$ $\mathit{\text{dominates}}$ $N[S]$. We say that $S$ $\mathit{\text{monitors}}$ vertices of $G$ as follows. Initially, all dominated vertices are monitored. This step is called the $\mathit{\text{domination}}$ step. Thereafter, the set of unmonitored vertices of which each is the only unmonitored neighbour of a monitored vertex, is monitored. This step is called a $\mathit{\text{propagation}}$ step and is repeated until the process terminates. The process terminates when the there are no monitored vertices which have exactly one unmonitored neighbour. This combined process of initial domination and subsequent propagation is called $\mathit{\text{power}}\mathit{\text{domination}}$. If all vertices of $G$ are monitored at termination, then $S$ is said to be a $\mathit{\text{power}}\mathit{\text{dominating}}\mathit{\text{set}}\mathit{\text{(PDS)}}\mathit{\text{of}}G$. The $\mathit{\text{power}}\mathit{\text{domination}}\mathit{\text{number}}\mathit{\text{of}}G$, denoted as ${\gamma }_p(G)$, is the minimum cardinality of a PDS of $G$. The $\mathit{\text{propagation}}\mathit{\text{radius}}\mathit{\text{of}}G$ is the minimum number of steps it takes a minimum PDS to monitor $V(G)$. In this paper we determine an upper bound on the propagation radius of $G$ with regards to power domination, in terms of $\delta$ and $n$. We show that this bound is only attained when ${\gamma }_p(G)=1$ and then improve this bound for ${\gamma }_p(G)\ge 2$. Sharpness examples for these bounds are provided. We also present sharp upper bounds on the propagation radius of split graphs. We present sharpness results for a known lower bound of the propagation radius for all $\mathrm{\Delta }\ge 3$.