1. On {k}-Roman graphsKenny Bešter Štorgel, Nina Chiarelli, Lara Fernández, J. Pascal Gollin, Claire Hilaire, Valeria Alejandra Leoni, Martin Milanič, 2025, objavljeni znanstveni prispevek na konferenci Opis: For a positive integer k, a {k}-Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, . . . , k} satisfying f (N(v)) ≥ k for each vertex v ∈ V with f (v) = 0. Every graph G satisfies γ{Rk}(G) ≤ 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 {k}-Roman graphs. This extends the concept of {k}-Roman trees studied by Wang et al. in 2021 to gen- eral graphs. We prove that for every k ≥ 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.
Ključne besede: graph domination, {k}-Roman domination, {k}-Roman graph, split graph, split join, NP-completeness
Objavljeno v RUP: 16.12.2025; Ogledov: 707; Prenosov: 3
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2. On the proper interval completion problem within some chordal subclassesFrançois Dross, Claire Hilaire, Ivo Koch, Valeria Alejandra Leoni, Nina Pardal, María Inés Lopez Pujato, Vinicius Fernandes dos Santos, 2025, izvirni znanstveni članek Opis: Given a property (graph class) Π, a graph G, and an integer k, the Π-completion problem consists of deciding whether we can turn G into a graph with the property Π by adding at most k edges to G. The Π-completion problem is known to be NP-hard for general graphs when Π is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.
Ključne besede: proper interval completion, split graph, threshold graph, quasi-threshold graph, caterpillar
Objavljeno v RUP: 06.08.2025; Ogledov: 869; Prenosov: 10
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