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Avoidable Vertices and Edges in Graphs

  • Jesse BeisegelEmail author
  • Maria Chudnovsky
  • Vladimir Gurvich
  • Martin Milanič
  • Mary Servatius
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11646)

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.

Notes

Acknowledgement

The authors are grateful to Ekkehard Köhler, Matjaž Krnc, Irena Penev, and Robert Scheffler for interest in their work and helpful remarks.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jesse Beisegel
    • 1
    Email author
  • Maria Chudnovsky
    • 2
  • Vladimir Gurvich
    • 3
  • Martin Milanič
    • 4
    • 5
  • Mary Servatius
    • 6
  1. 1.BTU Cottbus-SenftenbergCottbusGermany
  2. 2.Princeton UniversityPrincetonUSA
  3. 3.Higher School of EconomicsNational Research UniversityMoscowRussia
  4. 4.University of Primorska, IAMKoperSlovenia
  5. 5.University of Primorska, FAMNITKoperSlovenia
  6. 6.KoperSlovenia

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