p-Strong Roman Domination in Graphs
J. C. Valenzuela-Tripodoro, M. A. Mateos-Camacho, M. Cera, R. M. Casablanca, M. P. Álvarez-Ruiz
TL;DR
This work introduces the p-strong Roman domination parameter γ_{StR}^p(G), a relaxed variant of strong Roman domination with labels up to ceil((Δ+p)/p). It establishes NP-completeness of the associated p-StRD-number problem via an X3C-based reduction, derives general upper and lower bounds relating γ_{StR}^p(G) to classical domination parameters, and provides exact values for complete bipartite graphs and bi-stars, along with structural conditions for small γ_{StR}^p. A Robertson graph example demonstrates tightness of the bounds in a specific regular graph. Overall, the paper advances domination theory by introducing a tunable defense parameter and delivering both theoretical bounds and concrete exact values for key graph families, with potential applications in modeling robust defense against multiple simultaneous attacks.
Abstract
Domination in graphs is a widely studied field, where many different definitions have been introduced in the last years to respond to different network requirements. This paper presents a new dominating parameter based on the well-known strong Roman domination model. Given a positive integer $p$, we call a $p$-strong Roman domination function ($p$-StRDF) in a graph $G$ to a function $f:V(G)\rightarrow \{0,1,2, \ldots , \left\lceil \frac{Δ+p}{p} \right\rceil \}$ having the property that if $f(v)=0$, then there is a vertex $u\in N(v)$ such that $f(u) \ge 1+ \left\lceil \frac{ |B_0\cap N(u)|}{p} \right\rceil $, where $B_0$ is the set of vertices with label $0$. The $p$-strong Roman domination number $γ_{StR}^p(G)$ is the minimum weight (sum of labels) of a $p$-StRDF on $G$. We study the NP-completeness of the \emph{$p$-StRD}-problem, we also provide general and tight upper and lower bounds depending on several classical invariants of the graph and, finally, we determine the exact values for some families of graphs.