Longest Cycle Above Erdős-Gallai Bound

Authors Fedor V. Fomin , Petr A. Golovach , Danil Sagunov , Kirill Simonov



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Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Danil Sagunov
  • St. Petersburg Department of V.A. Steklov Institute of Mathematics, Russia
Kirill Simonov
  • Algorithms and Complexity Group, TU Wien, Austria

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Fedor V. Fomin, Petr A. Golovach, Danil Sagunov, and Kirill Simonov. Longest Cycle Above Erdős-Gallai Bound. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.55

Abstract

In 1959, Erdős and Gallai proved that every graph G with average vertex degree ad(G) ≥ 2 contains a cycle of length at least ad(G). We provide an algorithm that for k ≥ 0 in time 2^𝒪(k)⋅n^𝒪(1) decides whether a 2-connected n-vertex graph G contains a cycle of length at least ad(G)+k. This resolves an open problem explicitly mentioned in several papers. The main ingredients of our algorithm are new graph-theoretical results interesting on their own.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Longest path
  • longest cycle
  • fixed-parameter tractability
  • above guarantee parameterization
  • average degree
  • Erdős and Gallai theorem

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References

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