Avoidable vertices and edges in graphs

Jesse Beisegel, Maria Chudnovsky, Vladimir Gurvich, Martin Milanič, Mary Servatius

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

A vertex v in a graph G is said to be avoidable if every induced two-edge path with midpoint v is contained in an induced cycle. Generalizing Dirac’s theorem on the existence of simplicial vertices in chordal graphs, Ohtsuki et al. proved in 1976 that every graph has an avoidable vertex. In a different generalization, Chvátal et al. gave in 2002 a characterization of graphs without long induced cycles based on the concept of simplicial paths. We introduce the concept of avoidable induced paths as a common generalization of avoidable vertices and simplicial paths. We propose a conjecture that would unify the results of Ohtsuki et al. and of Chvátal et al. The conjecture states that every graph that has an induced k-vertex path also has an avoidable k-vertex path. We prove that every graph with an edge has an avoidable edge, thus establishing the case k= 2 of the conjecture. Furthermore, we point out a close relationship between avoidable vertices in a graph and its minimal triangulations and identify new algorithmic uses of avoidable vertices. More specifically, applying Lexicographic Breadth First Search and bisimplicial elimination orderings, we derive a polynomial-time algorithm for the maximum weight clique problem in a class of graphs generalizing the class of 1-perfectly orientable graphs and its subclasses chordal graphs and circular-arc graphs.

Original languageEnglish (US)
Title of host publicationAlgorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings
EditorsZachary Friggstad, Mohammad R. Salavatipour, Jörg-Rüdiger Sack
PublisherSpringer Verlag
Pages126-139
Number of pages14
ISBN (Print)9783030247652
DOIs
StatePublished - Jan 1 2019
Event16th International Symposium on Algorithms and Data Structures, WADS 2019 - Edmonton, Canada
Duration: Aug 5 2019Aug 7 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11646 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference16th International Symposium on Algorithms and Data Structures, WADS 2019
CountryCanada
CityEdmonton
Period8/5/198/7/19

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Beisegel, J., Chudnovsky, M., Gurvich, V., Milanič, M., & Servatius, M. (2019). Avoidable vertices and edges in graphs. In Z. Friggstad, M. R. Salavatipour, & J-R. Sack (Eds.), Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings (pp. 126-139). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11646 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-24766-9_10