Complexity Issues Concerning the Quadruple Roman Domination Problem in Graphs
V. S. R. Palagiri, G. P. Sharma, I. G. Yero
TL;DR
This paper investigates the computational complexity of the quadruple Roman domination problem (4RDP) by defining the 4RDF and the parameter $\gamma_{[4R]}(G)$, and analyzing its complexity across several graph classes. It establishes NP-completeness for star convex bipartite, comb convex bipartite, split, and planar graphs via ETC-based reductions, while proving polynomial solvability for threshold graphs. The authors derive bounds relating $\gamma_{[4R]}(G)$ to the classical domination number $\gamma(G)$, and present exact values for certain classes. They further address approximation limits, introduce a 4RDF approximation algorithm with a provable ratio, and prove APX-completeness for max-degree-4 graphs, complemented by an ILP formulation for exact optimization. The results illuminate the algorithmic landscape of 4RDP and point to practical approaches and future directions for heuristics and exact methods.
Abstract
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 $γ_{[4R]}(G)$. 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 $γ_{[4R]}(G)$ 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 $(1 - ε) \ln |V|$ for any $ε> 0$ 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.