arXiv subjects

Lara Fernández

Publications and source records attributed to Lara Fernández.

3 recordsLinked to original sources

arXiv2026

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

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 $\sum_{u\in N(v)} f(u) \geq k$ for each vertex $v\in V$ with $f (v) = 0$. Every graph $G$ satisfies $γ_{\{Rk\}}(G) \leq kγ(G)$, where $γ(G)$ is the domination number of $G$ and $γ_{\{Rk\}}(G)$ denotes the $\{k\}$-Roman domination number of $G$, that is, the minimum value of $\sum_{u\in V(G)} f(u)$ over all $\{k\}$-Roman dominating functions 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 $k\geq 2$, the problem of recognizing \hbox{$\{k\}$-Roman} graphs is \textsf{NP}-hard, even for split graphs. For ${k\geq 3}$, we give an alternative proof by generalizing several known results on domination in middle graphs to the hypergraph setting. Finally, we characterize the \kr property within two specific subclasses of split graphs: suns and their complements.

math.CO↗

arXiv2026

On $\{2\}$-Roman graph recognition of Partner Limited graphs

Given a graph $G=(V,E)$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex $u$ adjacent to $v$ with $f(u) = 2$, or two distinct vertices $x,\; y$ both adjacent to $v$ with $f(x)=f(y)=1$ (Chellali et al. 2016). Every graph $G$ satisfies $γ_{\{R2\}}(G) \leq 2γ(G)$, where $γ_{\{R2\}}(G)$ denotes the minimum weight of a $\{2\}$-Roman dominating function of $G$ and $γ(G)$ is the domination number of $G$. \emph{$\{2\}$-Roman graphs} are those for which the equality is reached (Klostermeyer et al. 2019). A characterization of $\{2\}$-Roman trees was given by Henning et al. in 2017. In 2025, Ferrari et al. characterized the $\{2\}$-Roman property by the existence of a minimum $\{2\}$-Roman dominating function of $G$ that assumes only $0, 2$-values. Afterwards in 2025, Bešter Štorgel et al. introduced the problem of recognizing $\{2\}$-Roman graphs, proved polinomiality for middle graphs, and characterized \hbox{$\{2\}$-Roman} split graphs that can be decomposed with respect to the split join operation into two smaller split graphs. Recognition complexity is still open for general graphs. In this paper we study the \hbox{$\{2\}$-Roman} property on graphs that can be decomposed into two smaller graphs with respect to the join and union operations, allowing to completely characterize the $\{2\}$-Roman property. The 4-path is the non trivial connected non $\{2\}$-Roman graph with the fewest number of vertices and edges. We classify the $\{2\}$-Roman property within specific families of non decomposable graphs with a limited number of 4-paths which are present in the decomposition of partner limited graphs; these are well-labelled spiders, the graphs in ZOO and some special split graphs.

math.CO↗

arXiv2026

Roman $\{2\}$-domination on Graphs with "few" 4-paths

Given a graph $G$ with vertex set $V$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} (or \emph{italian dominating function}) of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex $u$ adjacent to $v$ with $f(u) = 2$, or two distinct vertices $x,\; y$ both adjacent to $v$ with $f(x)=f(y)=1$. The decision problem associated with Roman $\{2\}$-domination is NP-complete even for bipartite graphs (Chellali et al., 2016). In this work we initiate the study of Roman $\{2\}$-domination on graph classes with a limited number of 4-paths. We base our study on a modular decomposition analysis. In particular, we study Roman $\{2\}$-domination under some operations in graphs such as join, union, complementation, addition of pendant vertices and addition of twin vertices. We then obtain the Roman $\{2\}$-domination number of spiders, well-labelled spiders and certain prime split graphs that are crucial in the modular decomposition of partner-limited graphs. In all, we provide linear-time algorithms to compute the Roman $\{2\}$-domination number of cographs, $P_4$-sparse graphs, $P_4$-tidy graphs and partner-limited graphs. Finally, we derive the NP-completeness of Roman $\{2\}$-domination on $P_4$-laden graphs.

math.CO↗