Bounds on the propagation radius in power domination

TL;DR

This work analyzes the propagation radius $rad_p(G)$ under power domination, establishing sharp upper bounds in terms of the graph order $n$ and minimum degree $\delta$, with equality cases when the power domination number $\gamma_p(G)=1$. It extends the bounds to scenarios with $\gamma_p(G)\ge 2$, introduces an excess-based framework to obtain refined bounds, and provides extremal constructions demonstrating sharpness. The paper also derives a sharp upper bound for split graphs in terms of $n$ and $\gamma_p(G)$, and proves a sharp lower bound on $rad_p$ in terms of $n$, the maximum degree $\Delta$, and $\gamma_p$, with explicit constructions for all feasible parameter sets. Together, these results illuminate how network monitoring speed (propagation radius) depends on degree, structure, and domination parameters, with implications for efficient PMU placement and related monitoring tasks.

Abstract

Let $G$ be a graph and let $S \subseteq V(G)$. It is said that $S$ \textit{dominates} $N[S]$. We say that $S$ \textit{monitors} vertices of $G$ as follows. Initially, all dominated vertices are monitored. If there exists a vertex $v \in G$ which is not monitored, but has all but one of its neighbours monitored, then $v$ becomes monitored itself. This step is called a \textit{propagation} step and is repeated until the process terminates. The process terminates when the there are no unmonitored vertices with exactly one unmonitored neighbour. This combined process of initial domination and subsequent propagation is called \textit{power domination}. If all vertices of $G$ are monitored at termination, then $S$ is said to be a \textit{power dominating set (PDS) of $G$}. The \textit{power domination number of $G$}, denoted as $γ_p(G)$, is the minimum cardinality of a PDS of $G$. The \textit{propagation radius 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 $δ$ and $n$. We show that this bound is only attained when $γ_p(G)=1$ and then improve this bound for $γ_p(G)\geq 2$. 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 $Δ\geq 3$.

Figures (10)

• Figure 1: A graph $G$ with $\{a,b\}$ being both a PDS and ZFS.
• Figure 2: The graph $G(2,4)$.
• Figure 3: The graph $G(3,4)$.
• Figure 4: The graph $G(4,5)$. Vertices $u,v$ are on the left, and $x,y$ on the right.
• Figure 5: A regular graph $G(4)$ attaining the $n-\delta - 1$ bound. The graph has $\delta = 4$ and $n=7$.

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4: zf_bar_2010
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4