Complexity Issues Concerning the Quadruple Roman Domination Problem in Graphs

Given a graph $G$ with vertex set $V(G)$, a mapping $h : V(G) \rightarrow \lbrace 0, 1, 2, 3, 4, 5 \rbrace$ is called a quadruple Roman dominating function (4RDF) for $G$ if it holds the following. Every vertex $x$ such that $h(x)\in \{0,1,2, 3\}$ satisfies that $h(N[x]) = \sum_{v\in N[x]} h(v) \geq |\{y:y \in N(x) \; \text{and} \; h(y) \neq 0\}|+4$, where $N(x)$ and $N[x]$ stands for the open and closed neighborhood of $x$, respectively. The smallest possible weight $\sum_{x \in V(G)} h(x)$ among all possible 4RDFs $h$ for $G$ is the quadruple Roman domination number of $G$, denoted by $\gamma_{[4R]}(G)$. This work is focused on complexity aspects for the problem of computing the value of this parameter for several graph classes. Specifically, it is shown that the decision problem concerning $\gamma_{[4R]}(G)$ is NP-complete when restricted to star convex bipartite, comb convex bipartite, split and planar graphs. In contrast, it is also proved that such problem can be efficiently solved for threshold graphs where an exact solution is demonstrated, while for graphs having an efficient dominating set, tight upper and lower bounds in terms of the classical domination number are given. In addition, some approximation results to the problem are given. That is, we show that the problem cannot be approximated within $(1 - \epsilon) \ln |V|$ for any $\epsilon > 0$ unless $P=NP$. An approximation algorithm for it is proposed, and its APX-completeness proved, whether graphs of maximum degree four are considered. Finally, an integer linear programming formulation for our problem is presented.