# 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)

## Abstract

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.

## References

1. 1.
Adamaszek, A., Adamaszek, M.: Uniqueness of graph square roots of girth six. Electron. J. Comb. 18(P139), 1 (2011)
2. 2.
Agnarsson, G., Halldórsson, M.M.: Coloring powers of planar graphs. SIAM J. Discrete Math. 16(4), 651–662 (2003)
3. 3.
Aigner, M., Fromme, M.: A game of cops and robbers. Discrete Appl. Math. 8(1), 1–12 (1984)
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)
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)
6. 6.
Bonamy, M., Lévêque, B., Pinlou, A.: 2-distance coloring of sparse graphs. J. Graph Theory 77(3), 190–218 (2014)
7. 7.
Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics. Springer, London (2008)
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.
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. 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)
11. 11.
Ducoffe, G.: Finding cut-vertices in the square roots of a graph. Technical report hal-01477981, UCA, Inria, CNRS, I3S, France (2017). https://hal.archives-ouvertes.fr/hal-01477981
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)
15. 15.
Fleischner, H.: The square of every two-connected graph is Hamiltonian. J. Comb. Theory Ser. B 16(1), 29–34 (1974)
16. 16.
Gallai, T.: Graphen mit triangulierbaren ungeraden Vielecken. Magyar Tud. Akad. Mat. Kutató Int. Közl 7, 3–36 (1962)
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:. 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:
20. 20.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, vol. 57. Elsevier, Amsterdam (2004)
21. 21.
Golumbic, M.C., Hammer, P.L.: Stability in circular arc graphs. J. Algorithms 9(3), 314–320 (1988)
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)
23. 23.
Heggernes, P.: Minimal triangulations of graphs: a survey. Discrete Math. 306(3), 297–317 (2006)
24. 24.
Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)
25. 25.
Lau, L.C.: Bipartite roots of graphs. ACM Trans. Algorithms (TALG) 2(2), 178–208 (2006)
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)
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)
28. 28.
Le, V.B., Tuy, N.N.: The square of a block graph. Discrete Math. 310(4), 734–741 (2010)
29. 29.
Leimer, H.-G.: Optimal decomposition by clique separators. Discrete Math. 113(1–3), 99–123 (1993)
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)
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)
32. 32.
Lin, Y.-L., Skiena, S.S.: Algorithms for square roots of graphs. SIAM J. Discrete Math. 8(1), 99–118 (1995)
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)
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)
36. 36.
Motwani, R., Sudan, M.: Computing roots of graphs is hard. Discrete Appl. Math. 54(1), 81–88 (1994)
37. 37.
Mukhopadhyay, A.: The square root of a graph. J. Comb. Theory 2(3), 290–295 (1967)
38. 38.
Nestoridis, N.V., Thilikos, D.M.: Square roots of minor closed graph classes. Discrete Appl. Math. 168, 34–39 (2014)
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: 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)
41. 41.
Randerath, B., Volkmann, L.: A characterization of well covered block-cactus graphs. Australas. J. Comb. 9, 307–314 (1994)
42. 42.
Robertson, N., Seymour, P.: Graph minors. II. algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)
43. 43.
Ross, I.C., Harary, F.: The square of a tree. Bell Syst. Tech. J. 39(3), 641–647 (1960)
44. 44.
Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55(2), 221–232 (1985)
45. 45.
Tucker, A.: Characterizing circular-arc graphs. Bull. Am. Math. Soc. 76(6), 1257–1260 (1970)
46. 46.
Wegner, G.: Graphs with given diameter and a coloring problem. University of Dortmund, Technical report (1977)Google Scholar