Finding Cut-Vertices in the Square Roots of a Graph

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)


The square of a given graph \(H=(V,E)\) is obtained from H by adding an edge between every two vertices at distance two in H. Given a graph class \(\mathcal {H}\), the \(\mathcal {H}\)-Square Root problem asks for the recognition of the squares of graphs in \(\mathcal {H}\). In this paper, we answer positively to an open question of [Golovach et al. IWOCA 2016] by showing that the squares of cactus-block graphs can be recognized in polynomial time. Our proof is based on new relationships between the decomposition of a graph by cut-vertices and the decomposition of its square by clique cutsets. More precisely, we prove that the closed neighbourhoods of cut-vertices in H induce maximal subgraphs of \(G = H^2\) with no clique-cutset. Furthermore, based on this relationship, we can compute from a given graph G the block-cut tree of a desired square root (if any). Although the latter tree is not uniquely defined, we show surprisingly that it can only differ marginally between two different roots. Our approach not only gives the first polynomial-time algorithm for the \(\mathcal {H}\)-Square Root problem for several graph classes \(\mathcal {H}\), but it also provides a unifying framework for the recognition of the squares of trees, block graphs and cactus graphs—among others.


  1. 1.
    Adamaszek, A., Adamaszek, M.: Uniqueness of graph square roots of girth six. Electron. J. Comb. 18(P139), 1 (2011)MathSciNetMATHGoogle Scholar
  2. 2.
    Agnarsson, G., Halldórsson, M.M.: Coloring powers of planar graphs. SIAM J. Discrete Math. 16(4), 651–662 (2003)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Aigner, M., Fromme, M.: A game of cops and robbers. Discrete Appl. Math. 8(1), 1–12 (1984)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Berry, A., Pogorelcnik, R., Simonet, G.: Organizing the atoms of the clique separator decomposition into an atom tree. Discrete Appl. Math. 177, 1–13 (2014)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bodlaender, H.-L., Kratsch, S., Kreuzen, V., Kwon, O.-J., Ok, S.: Characterizing width two for variants of treewidth. Discrete Appl. Math. 216(Part 1), 29–46 (2017)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bonamy, M., Lévêque, B., Pinlou, A.: 2-distance coloring of sparse graphs. J. Graph Theory 77(3), 190–218 (2014)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics. Springer, London (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Bouchitté, V., Todinca, I.: Treewidth and minimum fill-in: grouping the minimal separators. SIAM J. Comput. 31(1), 212–232 (2001)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Chang, M.-S., Ko, M.-T., Lu, H.-I.: Linear-time algorithms for tree root problems. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 411–422. Springer, Heidelberg (2006). doi: 10.1007/11785293_38 CrossRefGoogle Scholar
  10. 10.
    Cochefert, M., Couturier, J.-F., Golovach, P.A., Kratsch, D., Paulusma, D.: Parameterized algorithms for finding square roots. Algorithmica 74(2), 602–629 (2016)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Ducoffe, G.: Finding cut-vertices in the square roots of a graph. Technical report hal-01477981, UCA, Inria, CNRS, I3S, France (2017).
  12. 12.
    Ducoffe, G., Coudert, D.: Clique-decomposition revisited. In: Revision (Research Report on HAL, hal-01266147) (2017)Google Scholar
  13. 13.
    Farzad, B., Karimi, M.: Square-root finding problem in graphs, a complete dichotomy theorem. Technical report, arXiv arXiv:1210.7684 (2012)
  14. 14.
    Farzad, B., Lau, L.C., Tuy, N.N.: Complexity of finding graph roots with girth conditions. Algorithmica 62(1–2), 38–53 (2012)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Fleischner, H.: The square of every two-connected graph is Hamiltonian. J. Comb. Theory Ser. B 16(1), 29–34 (1974)CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Gallai, T.: Graphen mit triangulierbaren ungeraden Vielecken. Magyar Tud. Akad. Mat. Kutató Int. Közl 7, 3–36 (1962)MathSciNetMATHGoogle Scholar
  17. 17.
    Golovach, P., Heggernes, P., Kratsch, D., Lima, P., Paulusma, D.: Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2. In: Bodlaender, H.L., Woeginger, G.J. (eds.) WG 2017. LNCS, vol. 10520, pp. 275–288. Springer, Cham (2017). doi: 10.1007/978-3-319-68705-6_z. arXiv:1703.05102
  18. 18.
    Golovach, P., Kratsch, D., Paulusma, D., Stewart, A.: A linear kernel for finding square roots of almost planar graphs. In: SWAT, pp. 4:1–4:14 (2016)Google Scholar
  19. 19.
    Golovach, P.A., Kratsch, D., Paulusma, D., Stewart, A.: Finding cactus roots in polynomial time. In: Mäkinen, V., Puglisi, S.J., Salmela, L. (eds.) IWOCA 2016. LNCS, vol. 9843, pp. 361–372. Springer, Cham (2016). doi: 10.1007/978-3-319-44543-4_28 CrossRefGoogle Scholar
  20. 20.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  21. 21.
    Golumbic, M.C., Hammer, P.L.: Stability in circular arc graphs. J. Algorithms 9(3), 314–320 (1988)CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Harary, F., Karp, R.M., Tutte, W.T.: A criterion for planarity of the square of a graph. J. Comb. Theory 2(4), 395–405 (1967)CrossRefMathSciNetMATHGoogle Scholar
  23. 23.
    Heggernes, P.: Minimal triangulations of graphs: a survey. Discrete Math. 306(3), 297–317 (2006)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)CrossRefGoogle Scholar
  25. 25.
    Lau, L.C.: Bipartite roots of graphs. ACM Trans. Algorithms (TALG) 2(2), 178–208 (2006)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Lau, L.C., Corneil, D.G.: Recognizing powers of proper interval, split, and chordal graphs. SIAM J. Discrete Math. 18(1), 83–102 (2004)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Le, V.B., Nguyen, N.T.: A good characterization of squares of strongly chordal split graphs. Inf. Process. Lett. 111(3), 120–123 (2011)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Le, V.B., Tuy, N.N.: The square of a block graph. Discrete Math. 310(4), 734–741 (2010)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Leimer, H.-G.: Optimal decomposition by clique separators. Discrete Math. 113(1–3), 99–123 (1993)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Lih, K.-W., Wang, W.-F., Zhu, X.: Coloring the square of a \({K}_4\)-minor free graph. Discrete Math. 269(1), 303–309 (2003)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Lin, M.C., Rautenbach, D., Soulignac, F.J., Szwarcfiter, J.L.: Powers of cycles, powers of paths, and distance graphs. Discrete Appl. Math. 159(7), 621–627 (2011)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Lin, Y.-L., Skiena, S.S.: Algorithms for square roots of graphs. SIAM J. Discrete Math. 8(1), 99–118 (1995)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Lloyd, E., Ramanathan, S.: On the complexity of distance-2 coloring. In: ICCI, pp. 71–74. IEEE (1992)Google Scholar
  34. 34.
    Milanič, M., Schaudt, O.: Computing square roots of trivially perfect and threshold graphs. Discrete Appl. Math. 161(10), 1538–1545 (2013)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Molloy, M., Salavatipour, M.R.: A bound on the chromatic number of the square of a planar graph. J. Comb. Theory Ser. B 94(2), 189–213 (2005)CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Motwani, R., Sudan, M.: Computing roots of graphs is hard. Discrete Appl. Math. 54(1), 81–88 (1994)CrossRefMathSciNetMATHGoogle Scholar
  37. 37.
    Mukhopadhyay, A.: The square root of a graph. J. Comb. Theory 2(3), 290–295 (1967)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Nestoridis, N.V., Thilikos, D.M.: Square roots of minor closed graph classes. Discrete Appl. Math. 168, 34–39 (2014)CrossRefMathSciNetMATHGoogle Scholar
  39. 39.
    Le, V.B., Oversberg, A., Schaudt, O.: Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 360–371. Springer, Cham (2014). doi: 10.1007/978-3-319-12340-0_30 Google Scholar
  40. 40.
    Parra, A., Scheffler, P.: Characterizations and algorithmic applications of chordal graph embeddings. Discrete Appl. Math. 79(1–3), 171–188 (1997)CrossRefMathSciNetMATHGoogle Scholar
  41. 41.
    Randerath, B., Volkmann, L.: A characterization of well covered block-cactus graphs. Australas. J. Comb. 9, 307–314 (1994)MathSciNetMATHGoogle Scholar
  42. 42.
    Robertson, N., Seymour, P.: Graph minors. II. algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)CrossRefMathSciNetMATHGoogle Scholar
  43. 43.
    Ross, I.C., Harary, F.: The square of a tree. Bell Syst. Tech. J. 39(3), 641–647 (1960)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55(2), 221–232 (1985)CrossRefMathSciNetMATHGoogle Scholar
  45. 45.
    Tucker, A.: Characterizing circular-arc graphs. Bull. Am. Math. Soc. 76(6), 1257–1260 (1970)CrossRefMathSciNetMATHGoogle Scholar
  46. 46.
    Wegner, G.: Graphs with given diameter and a coloring problem. University of Dortmund, Technical report (1977)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université Côte d’Azur, Inria, CNRS, I3SSophia AntipolisFrance
  2. 2.National Institute for Research and Development in InformaticsBucharestRomania

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