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Obstructions for Minor-Closed Classes of Limiting Densities Below 3/2

Authors Antonios Kominatos , Reem Mahmoud , Dimitrios M. Thilikos

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph Minors
  • Limiting density
  • Obstruction set
  • Class property
  • Parametric graph

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Abstract

Given a graph class 𝒢, the limiting density of 𝒢 is defined as δ(𝒢) = lim_{n → ∞} ex(𝒢,n)/n where ex(𝒢,n) is the maximum number of edges of a graph in 𝒢 on n vertices. The limiting density δ(𝒢) is known to be a rational number when 𝒢 is a minor-closed graph class. For every δ ∈ [0,3/2), we prove that the set of ⊆-minimal minor-closed graph classes with densities > δ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs 𝒵, of total size n, either outputs the value of δ(excl(𝒵)) or reports that δ(excl(𝒵)) ≥ 3/2, where excl(𝒵) is the class of graphs excluding the graphs in 𝒵 as minors. The algorithm runs in 2^{poly(n)} time.

Cite As Get BibTex

Antonios Kominatos, Reem Mahmoud, and Dimitrios M. Thilikos. Obstructions for Minor-Closed Classes of Limiting Densities Below 3/2. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.29

Author Details

Antonios Kominatos
  • Department of Mathematics, National and Kapodistrian University of Athens, Greece
  • Inter-Institutional Graduate Program "Algorithms, Logic, and Discrete Mathematics" (ALMA), Athens, Greece
Reem Mahmoud
  • NYUAD, CS-program, Abu Dhabi, UAE
Dimitrios M. Thilikos
  • LIRMM, Univ Montpellier, CNRS, Montpellier, France

Funding

  • Kominatos, Antonios: Supported by the Scientific Research Network "Graphs, Association schemes and Geometries: structures, algorithms and computation" (grant no. W003324N) of the Research Foundation - Flanders (Belgium).
  • Thilikos, Dimitrios M.: Supported by the Franco-Norwegian project PHC AURORA 2024-25 and the French National Research Agency (ANR) under project GODASse ANR-24-CE48-4377 and under the France 2030 grant reference number ANR-24-RRII-0002 operated by the Inria Quadrant Program.

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