WADS 2019: Algorithms and Data Structures pp 126-139

# Avoidable Vertices and Edges in Graphs

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.

## References

1. 1.
Aboulker, P., Charbit, P., Trotignon, N., Vušković, K.: Vertex elimination orderings for hereditary graph classes. Discrete Math. 338(5), 825–834 (2015)
2. 2.
Addario-Berry, L., Chudnovsky, M., Havet, F., Reed, B., Seymour, P.: Bisimplicial vertices in even-hole-free graphs. J. Comb. Theory Ser. B 98(6), 1119–1164 (2008)
3. 3.
Bang-Jensen, J., Huang, J., Prisner, E.: In-tournament digraphs. J. Comb. Theory Ser. B 59(2), 267–287 (1993)
4. 4.
Barot, M., Geiss, C., Zelevinsky, A.: Cluster algebras of finite type and positive symmetrizable matrices. J. Lond. Math. Soc. 73(3), 545–564 (2006)
5. 5.
Berry, A., Blair, J.R.S., Bordat, J.-P., Simonet, G.: Graph extremities defined by search algorithms. Algorithms 3(2), 100–124 (2010)
6. 6.
Berry, A., Blair, J.R.S., Heggernes, P., Peyton, B.W.: Maximum cardinality search for computing minimal triangulations of graphs. Algorithmica 39(4), 287–298 (2004)
7. 7.
Berry, A., Bordat, J.-P.: Separability generalizes Dirac’s theorem. Discrete Appl. Math. 84(1–3), 43–53 (1998)
8. 8.
Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: grouping the minimal separators. SIAM J. Comput. 31(1), 212–232 (2001)
9. 9.
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)
10. 10.
Brešar, B., Hartinger, T.R., Kos, T., Milanič, M.: 1-perfectly orientable $$K_4$$-minor-free and outerplanar graphs. Discrete Appl. Math. 248, 33–45 (2018)
11. 11.
Chvátal, V., Rusu, I., Sritharan, R.: Dirac-type characterizations of graphs without long chordless cycles. Discrete Math. 256(1–2), 445–448 (2002)
12. 12.
Dirac, G.A.: On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)
13. 13.
Farber, M., Jamison, R.E.: Convexity in graphs and hypergraphs. SIAM J. Algebraic Discrete Methods 7(3), 433–444 (1986)
14. 14.
Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pac. J. Math. 15, 835–855 (1965)
15. 15.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Annals of Discrete Mathematics, vol. 57, 2nd edn. Elsevier Science B.V., Amsterdam (2004)
16. 16.
Habib, M., McConnell, R., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoret. Comput. Sci. 234(1–2), 59–84 (2000)
17. 17.
Hartinger, T.R., Milanič, M.: Partial characterizations of $$1$$-perfectly orientable graphs. J. Graph Theory 85(2), 378–394 (2017)
18. 18.
Hartinger, T.R., Milanič, M.: 1-perfectly orientable graphs and graph products. Discrete Math. 340(7), 1727–1737 (2017)
19. 19.
Heggernes, P.: Minimal triangulations of graphs: a survey. Discrete Math. 306(3), 297–317 (2006)
20. 20.
Kammer, F., Tholey, T.: Approximation algorithms for intersection graphs. Algorithmica 68(2), 312–336 (2014)
21. 21.
Ohtsuki, T., Cheung, L.K., Fujisawa, T.: Minimal triangulation of a graph and optimal pivoting order in a sparse matrix. J. Math. Anal. Appl. 54(3), 622–633 (1976)
22. 22.
Rose, D.J.: Symmetric elimination on sparse positive definite systems and the potential flow network problem. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), Harvard University (1970)Google Scholar
23. 23.
Rose, D.J.: Triangulated graphs and the elimination process. J. Math. Anal. Appl. 32, 597–609 (1970)
24. 24.
Rose, D.J.: A Graph-Theoretic Study of the Numerical Solution of Sparse Positive Definite Systems of Linear Equations, pp. 183–217. Academic Press, New York (1972)Google Scholar
25. 25.
Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)
26. 26.
Skrien, D.J.: A relationship between triangulated graphs, comparability graphs, proper interval graphs, proper circular-arc graphs, and nested interval graphs. J. Graph Theory 6(3), 309–316 (1982)
27. 27.
Spinrad, J.P.: Efficient Graph Representations. Fields Institute Monographs, vol. 19. American Mathematical Society, Providence (2003)
28. 28.
Urrutia, J., Gavril, F.: An algorithm for fraternal orientation of graphs. Inf. Process. Lett. 41(5), 271–274 (1992)
29. 29.
Voloshin, V.I.: Properties of triangulated graphs. In: Operations Research and Programming, pp. 24–32. Shtiintsa, Kishinev (1982)Google Scholar
30. 30.
West, D.B.: Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)
31. 31.
Ye, Y., Borodin, A.: Elimination graphs. ACM Trans. Algorithms 8(2), 23 (2012). Art. 14

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Jesse Beisegel
• 1
Email author
• Maria Chudnovsky
• 2
• 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

## Personalised recommendations

### Citepaper 