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Lookup NU author(s): Dr Konrad DabrowskiORCiD
This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).
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.
Author(s): Dabrowski KK, Lozin V, Milanič M, Munaro A, Paulusma D, Zamaraev V
Editor(s): Jan Goedgebeur and Paweł Rzążewski
Publication type: Conference Proceedings (inc. Abstract)
Publication status: Published
Conference Name: 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)
Year of Conference: 2026
Pages: 14:1-14:15
Online publication date: 02/07/2026
Acceptance date: 03/04/2026
Date deposited: 06/05/2026
Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
URL: https://doi.org/10.4230/LIPIcs.WG.2026.14
DOI: 10.4230/LIPIcs.WG.2026.14
Notes: Conference site: https://wg2026.be/
Series Title: Leibniz International Proceedings in Informatics (LIPIcs)
