p-Strong Roman Domination in Graphs

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 $f:V(G)\to \{0,1,2,\dots ,\lceil \frac{\mathrm{\Delta }+p}{p}\rceil \}$ having the property that if $f(v)=0$, then there is a vertex $u\in N(v)$ such that $f(u)\ge 1+\lceil \frac{\mathrm{\mid }{B}_0\cap N(u)\mathrm{\mid }}{p}\rceil$, where $B_0$ is the set of vertices with label $0$. The $p$-strong Roman domination number ${\gamma }_{StR}^p(G)$ 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.