Transversals of Longest Cycles in Chordal and Bounded Tree-Width Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

Let \(\mathrm {lct}(G)\) be the minimum size of a set of vertices that intersects every longest cycle of a 2-connected graph G. Let \(\mathrm {tw}(G)\) be the tree-width of G and \(\omega (G)\) be the size of a maximum clique in G. We show that \(\mathrm {lct}(G)\le \mathrm {tw}(G)-1\) for every G, and that \(\mathrm {lct}(G)\le \max \{1, {\omega (G){-}3}\}\) if G is chordal. Those results imply as corollaries that all longest cycles intersect in 2-connected series-parallel graphs and in 3-trees. We also strengthen the latter result and show that all longest cycles intersect in 2-connected graphs of tree-width at most 3, also known as partial 3-trees.

References

  1. 1.
    Grötschel, M.: On intersections of longest cycles. In: Bollobás, B. (ed.) Graph Theory and Combinatorics, pp. 171–189 (1984)Google Scholar
  2. 2.
    Thomassen, C.: Hypohamiltonian graphs and digraphs. In: Alavi, Y., Lick, D.R. (eds.) Theory and Applications of Graphs. LNM, vol. 642, pp. 557–571. Springer, Heidelberg (1978).  https://doi.org/10.1007/BFb0070410 CrossRefGoogle Scholar
  3. 3.
    Rautenbach, D., Sereni, J.S.: Transversals of longest paths and cycles. SIAM J. Discret. Math. 28(1), 335–341 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Jobson, A., Kzdy, A., Lehel, J., White, S.: Detour trees. Discret. Appl. Math. 206, 73–80 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    van Aardt, S.A., Burger, A.P., Dunbar, J.E., Frick, M., Llano, B., Thomassen, C., Zuazua, R.: Destroying longest cycles in graphs and digraphs. Discret. Appl. Math. 186(Suppl. C), 251–259 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fernandes, C., Gutiérrez, J.: Hitting all longest cycles in a graph. In: Anais do XXXVII congresso da sociedade brasileira de computação, pp. 87–90 (2017)Google Scholar
  7. 7.
    Balister, P., Győri, E., Lehel, J., Schelp, R.: Longest paths in circular arc graphs. Comb. Probab. Comput. 13(3), 311–317 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cerioli, M., Lima, P.: Intersection of longest paths in graph classes. Electron. Notes Discret. Math. 55, 139–142 (2016)CrossRefMATHGoogle Scholar
  9. 9.
    Chen, F.: Nonempty intersection of longest paths in a graph with a small matching number. Czechoslov. Math. J. 65(140), 545–553 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chen, G., Ehrenmüller, J., Fernandes, C., Heise, C., Shan, S., Yang, P., Yates, A.: Nonempty intersection of longest paths in seriesparallel graphs. Discret. Math. 340(3), 287–304 (2017)CrossRefMATHGoogle Scholar
  11. 11.
    de Rezende, S., Fernandes, C., Martin, D., Wakabayashi, Y.: Intersecting longest paths. Discret. Math. 313, 1401–1408 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Golan, G., Shan, S.: Nonempty intersection of longest paths in \(2{K_2}\)-free graphs. https://arxiv.org/abs/1611.05967 (2016)
  13. 13.
    Klavžar, S., Petkovšek, M.: Graphs with nonempty intersection of longest paths. Ars Comb. 29, 43–52 (1990)MathSciNetMATHGoogle Scholar
  14. 14.
    Zamfirescu, T.: On longest paths and circuits in graphs. Math. Scand. 38(2), 211–239 (1976)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cerioli, M.R., Fernandes, C.G., Gómez, R., Gutiérrez, J., Lima, P.T.: Transversals of longest paths. Electron. Notes Discret. Math. 62(Suppl. C), 135–140 (2017). IX Latin and American Algorithms, Graphs and Optimization, LAGOS 2017CrossRefGoogle Scholar
  16. 16.
    Cerioli, M.R., Fernandes, C.G., Gómez, R., Gutiérrez, J., Lima, P.T.: Transversals of longest paths (2017). https://arxiv.org/abs/1712.07086
  17. 17.
    Chen, G., Faudree, R.J., Gould, R.J.: Intersections of longest cycles in k-connected graphs. J. Comb. Theory Ser. B 72(1), 143–149 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hippchen, T.: Intersections of longest paths and cycles. Ph.D. thesis, Georgia State University (2008)Google Scholar
  19. 19.
    Jendrol, S., Skupień, Z.: Exact numbers of longest cycles with empty intersection. Eur. J. Comb. 18(5), 575–578 (1997)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Stewart, I.A., Thompson, B.: On the intersections of longest cycles in a graph. Exp. Math. 4(1), 41–48 (1995)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Diestel, R.: Graph Theory. GTM, vol. 173, 4th edn. Springer, Heidelberg (2017)CrossRefMATHGoogle Scholar
  22. 22.
    Bodlaender, H.: A partial \(k\)-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1), 1–45 (1998)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Gavril, F.: The intersection graphs of subtrees in trees are exactly the chordal graphs. J. Comb. Theory Ser. B 16(1), 47–56 (1974)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Heinz, M.: Tree-decomposition: graph minor theory and algorithmic implications. Master’s thesis, Technischen Universität Wien (2013)Google Scholar
  25. 25.
    Fomin, F., Thilikos, D.: New upper bounds on the decomposability of planar graphs. J. Graph Theory 51(1), 53–81 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Alon, N., Seymour, P., Thomas, R.: A separator theorem for nonplanar graphs. J. Am. Math. Soc. 3, 801–808 (1990)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Shabbir, A., Zamfirescu, C., Zamfirescu, T.: Intersecting longest paths and longest cycles: a survey. Electron. J. Graph Theory Appl. 1, 56–76 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Ciência da ComputaçãoUniversidade de São PauloSão PauloBrazil

Personalised recommendations