On $\{k\}$-Roman graphs: complexity of recognition and the case of split graphs

TL;DR

The paper addresses the problem of recognizing $\{k\}$-Roman graphs, i.e., graphs where $\gamma_{\{Rk\}}(G)=k\gamma(G)$, and introduces a hypergraph compatibility framework that links $\{k\}$-Roman membership to perfect matchings in $k$-uniform hypergraphs. Using this framework, the authors prove that, for every $k\ge3$, recognizing $\{k\}$-Roman graphs is NP-hard, even when restricted to split graphs, via a reduction from exact cover by $k$-sets and the middle-graph connection. For $k=2$, they provide partial structural results on split graphs, notably a linear-time reduction to split-join decompositions and complete classifications for the prime families suns and cosuns. The results offer both hardness foundations and structural insights, guiding future work on tractable cases and detailed characterizations within split graphs. These advances deepen the understanding of the range and limits of $\{k\}$-Roman domination across graph families.

Abstract

For a positive integer $k$, a $\{k\}$-Roman dominating function of a graph $G = (V,E)$ is a function $f\colon V \rightarrow \{0,1,\ldots,k\}$ satisfying $f (N(v)) \geq k$ for each vertex $v\in V$ with $f (v) = 0$. Every graph $G$ satisfies $γ_{\{Rk\}}(G) \leq kγ(G)$, where $γ_{\{Rk\}}(G)$ denotes the minimum weight of a $\{k\}$-Roman dominating function of $G$ and $γ(G)$ is the domination number of $G$. In this work we study graphs for which the equality is reached, called \emph{$\{k\}$-Roman graphs}. This extends the concept of $\{k\}$-Roman trees studied by Wang et al. in 2021 to general graphs. We prove that for every $k\geq 3$, the problem of recognizing $\{k\}$-Roman graphs is NP-hard, even when restricted to split graphs. We provide partial answers to the question of which split graphs are $\{2\}$-Roman: we characterize $\{2\}$-Roman split graphs that can be decomposed with respect to the split join operation into two smaller split graphs and classify the $\{k\}$-Roman property within two specific families of split graphs that are prime with respect to the split join operation: suns and their complements.

Figures (2)

• Figure 1: Examples of the $3$-sun (left) and the $4$-sun (right).
• Figure 2: The complement of the $5$-sun with vertices $v_1,v_2\in K$ and $x,y_1,y_2\in I$, where vertex $x$ is not adjacent to $v_1$ nor $v_2$, vertex $y_1$ is adjacent to $v_1$ but not to $v_2$, and vertex $y_2$ is adjacent to $v_2$ but not to $v_1$.

Theorems & Definitions (40)

• Theorem 2.1
• Corollary 2.2
• proof
• Definition 2.1
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof