arXiv subjects

Kenny Bešter Štorgel

Publications and source records attributed to Kenny Bešter Štorgel.

3 recordsLinked to original sources

arXiv2026

Awesome graph parameters

For a graph $G$, we denote by $α(G)$ the size of a maximum independent set and by $ω(G)$ the size of a maximum clique in $G$. Our paper lies on the edge of two lines of research, related to $α$ and $ω$, respectively. One of them studies $α$-variants of graph parameters, such as $α$-treewidth or $α$-degeneracy. The second line deals with graph classes where some parameters are bounded by a function of $ω(G)$. A famous example of this type is the family of $χ$-bounded classes, where the chromatic number $χ(G)$ is bounded by a function of $ω(G)$. A Ramsey-type argument implies that if the $α$-variant of a graph parameter $ρ$ is bounded by a constant in a hereditary class $\mathcal{G}$, then $ρ$ is bounded by a function of $ω$ in $\mathcal{G}$. If the reverse implication also holds, we say that $ρ$ is awesome. Otherwise, we say that $ρ$ is awful. In the present paper, we identify a number of awesome and awful graph parameters, derive some algorithmic applications of awesomeness, and propose a number of open problems related to these notions.

math.CO↗

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

Tree-independence number of $K_{1,d}$-free graph classes

In this paper, we investigate the tree-independence number of graph classes that do not contain $K_{1,d}$ as an induced subgraph. Dallard et al. conjectured that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. Our main contribution towards this conjecture is showing that the conjecture holds for outerstring graphs. Additionally we give linear and quadratic bounds for the tree-independence number of various $K_{1,d}$-free graph classes, sharpening previous bounds. Finally, we bound the tree-independence number of $K_{2,d}$-free graphs additionally forbidding holes of length at least $5$.

math.CO↗