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2. Graph classes closed under self-intersectionKonrad K. Dabrowski, Vadim V. Lozin, Martin Milanič, Andrea Munaro, Daniël Paulusma, Viktor Zamaraev, 2026, objavljeni znanstveni prispevek na konferenci Opis: A graph class is monotone if it is closed under taking subgraphs. A monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a tripod, i.e. a disjoint union of subdivided claws and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard graph problems admit polynomial-time algorithms. These dichotomies do not extend to the universe of all hereditary classes. This leads to the question of whether we can extend known dichotomies for monotone classes to larger families of hereditary classes. We answer this question affirmatively by considering the family of hereditary graph classes closed under self-intersection. This family is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. In contrast to monotone classes excluding a tripod, these classes do not necessarily have bounded treewidth; in fact, they do not even need to be sparse. We use our characterization to give a complete dichotomy for Maximum Independent Set, and its weighted variant, on self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. Our dichotomy generalizes several known results on Maximum Independent Set in the literature. We also apply our characterization to obtain a dichotomy for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions, and for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions. Finally, we use our characterization to obtain a dichotomy for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.
Ključne besede: graph classes, self-intersection closed, dichotomy, independent set, clique-width, treewidth
Objavljeno v RUP: 15.07.2026; Ogledov: 342; Prenosov: 8
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3. Excluding an induced wheel minor in graphs without large induced starsMujin Choi, Claire Hilaire, Martin Milanič, Sebastian Wiederrecht, 2026, objavljeni znanstveni prispevek na konferenci Opis: We study a conjecture due to Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht stating 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. A k-wheel is the graph obtained from a cycle of length k by adding a vertex adjacent to all vertices of the cycle. We show that the conjecture of Dallard et al. is true when H is a k-wheel for any k at least 3. Our proof uses a generalization of the concept of brambles to tree-independence number. As a consequence of our main result, several important NP-hard problems such as Maximum Independent Set are tractable on K_{1,d}-free graphs without large induced wheel minors. Moreover, for fixed d and k, we provide a polynomial-time algorithm that, given a K_{1,d}-free graph G as input, finds an induced minor model of a k-wheel in G if one exists.
Ključne besede: induced minor, wheel, tree-independence number, Maximum Independent Set
Objavljeno v RUP: 25.03.2026; Ogledov: 742; Prenosov: 2
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4. Paint cost spectrum of perfect k-ary treesSonwabile Mafunda, Jonathan L. Merzel, Katherine E. Perry, Anna Varvak, 2026, izvirni znanstveni članek Opis: We determine the paint cost spectrum for perfect k-ary trees. A coloring of the vertices of a graph G with d colors is said to be d-distinguishing if only the trivial automorphism preserves the color classes. The smallest such d is the distinguishing number of G and is denoted Dist(G). The paint cost of d-distinguishing G, denoted ρd(G), is the minimum size of the complement of a color class over all d-distinguishing colorings. A subset S of the vertices of G is said to be a fixing set for G if the only automorphsim that fixes the vertices in S pointwise is the trivial automorphism. The cardinality of a smallest fixing set is denoted Fix(G). In this paper, we explore the breaking of symmetry in perfect k-ary trees by investigating what we define as the paint cost spectrum of a graph G: (Dist(G); ρDist(G)(G), ρDist(G)+1(G), . . . , ρFix(G)+1(G)) and the paint cost ratio of G, which is defined to be the fraction of paint costs in the paint cost spectrum equal to Fix(G). We determine both the paint cost spectrum and the paint cost ratio completely for perfect k-ary trees. We also prove a lemma that is of interest in its own right: given an n-tuple, n ≥ 2 of distinct elements of an ordered abelian group and 1 ≤ k ≤ n! − 1, there exists a k × n row permuted matrix with distinct column sums.
Ključne besede: distinguishing coloring, fixing set, symmetry, cost of distinguishing
Objavljeno v RUP: 23.03.2026; Ogledov: 690; Prenosov: 12
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5. Mobile mutual-visibility sets in graphsMagda Dettlaff, Magdalena Lemańska, Juan A. Rodríguez-Velázquez, Ismael G. Yero, 2026, izvirni znanstveni članek Opis: Given a connected graph G, the mutual-visibility number of G is the cardinality of a largest set S such that for every pair of vertices x, y ∈ S there exists a shortest x, y-path whose interior vertices are not contained in S. Assume that a robot is assigned to each vertex of the set S. At each stage, one robot can move to a neighbouring vertex. Then S is a mobile mutual-visibility set of G if there exists a sequence of moves of the robots such that all the vertices of G are visited while maintaining the mutual-visibility property at all times. The mobile mutual-visibility number of G, denoted Mobµ(G), is the cardinality of a largest mobile mutual-visibility set of G. In this paper we introduce the concept of the mobile mutual-visibility number of a graph. We begin with some basic properties of the mobile mutual-visibility number of G and its relationship with the mutual-visibility number of G. We give exact values of Mobµ(G) for particular classes of graphs, i.e. cycles, wheels, complete bipartite graphs, and block graphs (in particular trees). Moreover, we present bounds for the lexicographic product of two graphs and show characterizations of the graphs achieving the limit values of some of these bounds. As a consequence of this study, we deduce that the decision problem concerning finding the mobile mutual-visibility number is NP-hard. Finally, we focus our attention on the mobile mutual-visibility number of line graphs of complete graphs, prism graphs and strong grids of two paths.
Ključne besede: mobile mutual-visibility set, mutual-visibility number, total mutual-visibility
Objavljeno v RUP: 03.03.2026; Ogledov: 653; Prenosov: 41
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8. Generalization of edge general position problemPaul Manuel, R. Prabha, Sandi Klavžar, 2025, izvirni znanstveni članek Opis: The edge geodesic cover problem of a graph G is to find a smallest number of geodesics that cover the edge set of G. The edge k-general position problem is introduced as the problem to find a largest set S of edges of G such that at most k-1 edges of S lie on a common geodesic. We show that these are dual min-max problems and connect them to an edge geodesic partition problem. Using these connections, exact values of the edge k-general position number is determined for different values of k and for various networks including torus networks, hypercubes, and Benes networks.
Ključne besede: general position set, edge geodesic cover problem, edge k-general position problem, torus network, hypercube, Benes network
Objavljeno v RUP: 03.11.2025; Ogledov: 809; Prenosov: 9
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9. The independence polynomial of trees is not always log-concave starting from order 26Ohr Kadrawi, Vadim Levit, 2025, izvirni znanstveni članek Opis: An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in G is represented by α(G). The independence polynomial of a graph G = (V, E) was introduced by Gutman and Harary in 1983 and is defined as
I(G; x) = Σ_{k = 0}^α(G) s_k x^k = s₀ + s₁x + s₂x² + ... + s_α(G)x^α(G), where sk represents the number of independent sets in G of size k.
The problem raised by Alavi, Malde, Schwenk, and Erdös in 1987 stated that the independence polynomials of trees are unimodal, and many researchers believed that this problem could be strengthened up to its corresponding log-concave version. However, in 2023, this conjecture was shown to be false by Kadrawi, Levit, Yosef, and Mizrachi. In this paper, we provide further evidence against this conjecture by presenting infinite families of trees with independence polynomials that are not log-concave.
Ključne besede: tree, independent set, independence polynomial, unimodality, log-concavity
Objavljeno v RUP: 22.10.2025; Ogledov: 1810; Prenosov: 8
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10. Mutual-visibility problems in Kneser and Johnson graphsGülnaz Boruzanlı Ekinci, Csilla Bujtás, 2025, izvirni znanstveni članek Opis: Let G be a connected graph and X ⊆ V(G). By definition, two vertices u and v are X-visible in G if there exists a shortest u, v-path with all internal vertices being outside of the set X. The largest size of X such that any two vertices of G (resp. any two vertices from X) are X-visible is the total mutual-visibility number (resp. the mutual-visibility number) of G.
In this paper, we determine the total mutual-visibility number of Kneser graphs, bipartite Kneser graphs, and Johnson graphs. The formulas proved for Kneser, and bipartite Kneser graphs are related to the size of transversal-critical uniform hypergraphs, while the total mutual-visibility number of Johnson graphs is equal to a hypergraph Turán number. Exact values or estimations for the mutual-visibility number over these graph classes are also established.
Ključne besede: mutual-visibility set, total mutual-visibility set, Kneser graph, bipartite Kneser graph, Johnson graph, Turán-type problem, covering design
Objavljeno v RUP: 22.10.2025; Ogledov: 1060; Prenosov: 12
Celotno besedilo (426,16 KB) |
