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Graph Classes Closed Under Self-Intersection

Authors Konrad K. Dabrowski , Vadim Lozin , Martin Milanič , Andrea Munaro , Daniël Paulusma , Viktor Zamaraev

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LIPIcs.WG.2026.14.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • graph classes
  • self-intersection closed
  • dichotomy
  • independent set
  • clique-width
  • treewidth

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Abstract

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.

Cite As Get BibTex

Konrad K. Dabrowski, Vadim Lozin, Martin Milanič, Andrea Munaro, Daniël Paulusma, and Viktor Zamaraev. Graph Classes Closed Under Self-Intersection. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.14

Author Details

Konrad K. Dabrowski
  • Newcastle University, Newcastle upon Tyne, UK
Vadim Lozin
  • University of Warwick, Coventry, UK
Martin Milanič
  • FAMNIT and IAM, University of Primorska, Koper, Slovenia
Andrea Munaro
  • University of Parma, Italy
Daniël Paulusma
  • Durham University, UK
Viktor Zamaraev
  • University of Liverpool, UK

Funding

  • Milanič, Martin: Supported in part by the Slovenian Research and Innovation Agency (I0-0035, research program P1-0285 and research projects J1-60012, J1-70035, J1-70046, and N1-0370) and by the research program CogniCom (0013103) at the University of Primorska.
  • Paulusma, Daniël: Supported by Leverhulme Trust Research Project Grant RPG-2024-182.

Acknowledgements

This work was supported by the International Centre for Mathematical Sciences.

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