Avoidable Vertices and Edges in Graphs
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 differ...
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| Published in | Algorithms and Data Structures Vol. 11646; pp. 126 - 139 |
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| Main Authors | , , , , |
| Format | Book Chapter |
| Language | English |
| Published |
Switzerland
Springer International Publishing AG
2019
Springer International Publishing |
| Series | Lecture Notes in Computer Science |
| Online Access | Get full text |
| ISBN | 3030247651 9783030247652 |
| ISSN | 0302-9743 1611-3349 |
| DOI | 10.1007/978-3-030-24766-9_10 |
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| Summary: | 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 $$k = 2$$ 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. |
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| Bibliography: | Part of the work for this paper was done in the framework of two bilateral projects between Germany and Slovenia, financed by DAAD and the Slovenian Research Agency (BI-DE/17-19-18 and BI-DE/19-20-04). The third named author was partially funded by Russian Academic Excellence Project ‘5-100’. The work of the fourth named author is supported in part by the Slovenian Research Agency (I0-0035, research program P1-0285 and research projects N1-0032, N1-0102, J1-7051, J1-9110). Part of the work was done while the fourth named author was visiting Osaka Prefecture University in Japan, under the operation Mobility of Slovene higher education teachers 2018–2021, co-financed by the Republic of Slovenia and the European Union under the European Social Fund. Original 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 2$$\end{document} 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. |
| ISBN: | 3030247651 9783030247652 |
| ISSN: | 0302-9743 1611-3349 |
| DOI: | 10.1007/978-3-030-24766-9_10 |