Abstract
Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.
Supported by the Research Council of Norway via the project “CLASSIS” and the Leverhulme Trust (RPG-2016-258).
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References
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 305–1317 (1996)
Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)
Cochefert, M., Couturier, J.-F., Golovach, P.A., Kratsch, D., Paulusma, D.: Sparse square roots. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 177–188. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-45043-3_16
Cochefert, M., Couturier, J., Golovach, P.A., Kratsch, D., Paulusma, D.: Parameterized algorithms for finding square roots. Algorithmica 74, 602–629 (2016)
Courcelle, B.: The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues. Informatique Théorique Appl. 26, 257–286 (1992)
Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138. Cambridge University Press, Cambridge (2012)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (2012)
Ducoffe G., Finding cut-vertices in the square roots of a graph. In: Proceedings of the WG 2017. LNCS (to Appear)
Farzad, B., Karimi, M.: Square-root finding problem in graphs, a complete dichotomy theorem. CoRR abs/1210.7684 (2012)
Farzad, B., Lau, L.C., Le, V.B., Tuy, N.N.: Complexity of finding graph roots with girth conditions. Algorithmica 62, 38–53 (2012)
Golovach, P.A., Heggernes, P., Kratsch, D., Lima, P.T., Paulusma, D.: Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2. CoRR abs/1703.05102 (2017)
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). https://doi.org/10.1007/978-3-319-44543-4_28
Golovach, P.A., Kratsch, D., Paulusma, D., Stewart, A.: A linear kernel for finding square roots of almost planar graphs. In: Proceedings of the 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016, vol. 53, pp. 4:1–4:14. Leibniz International Proceedings in Informatics (2016)
Golovach, P.A., Kratsch, D., Paulusma, D., Stewart, A.: Squares of low clique number. Electron. Notes Discrete Math. 55, 195–198 (2016). 14th Cologne Twente Workshop 2016, CTW 2016
Kinnersley, N.G., Langston, M.A.: Obstruction set isolation for the gate matrix layout problem. Discrete Appl. Math. 54(2–3), 169–213 (1994)
Lau, L.C.: Bipartite roots of graphs. ACM Trans. Algorithms 2, 178–208 (2006)
Lau, L.C., Corneil, D.G.: Recognizing powers of proper interval, split, and chordal graphs. SIAM J. Discrete Math. 18, 83–102 (2004)
Le, V.B., Oversberg, A., Schaudt, O.: Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs. Theoret. Comput. Sci. 602, 39–49 (2015)
Le, V.B., Oversberg, A., Schaudt, O.: A unified approach for recognizing squares of split graphs. Theoret. Comput. Sci. 648, 26–33 (2016)
Le, V.B., Tuy, N.N.: The square of a block graph. Discrete Math. 310, 734–741 (2010)
Le, V.B., Tuy, N.N.: A good characterization of squares of strongly chordal split graphs. Inf. Process. Lett. 111, 120–123 (2011)
Lin, Y.-L., Skiena, S.S.: Algorithms for square roots of graphs. In: Hsu, W.-L., Lee, R.C.T. (eds.) ISA 1991. LNCS, vol. 557, pp. 12–21. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-54945-5_44
Milanic, M., Oversberg, A., Schaudt, O.: A characterization of line graphs that are squares of graphs. Discrete Appl. Math. 173, 83–91 (2014)
Milanic, M., Schaudt, O.: Computing square roots of trivially perfect and threshold graphs. Discrete Appl. Math. 161, 1538–1545 (2013)
Motwani, R., Sudan, M.: Computing roots of graphs is hard. Discrete Appl. Math. 54, 81–88 (1994)
Mukhopadhyay, A.: The square root of a graph. J. Comb. Theory 2, 290–295 (1967)
Nestoridis, N.V., Thilikos, D.M.: Square roots of minor closed graph classes. Discrete Appl. Math. 168, 34–39 (2014)
Ross, I.C., Harary, F.: The square of a tree. Bell Syst. Tech. J. 39, 641–647 (1960)
Sysło, M.M.: Characterizations of outerplanar graphs. Discrete Math. 26(1), 47–53 (1979)
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Golovach, P.A., Heggernes, P., Kratsch, D., Lima, P.T., Paulusma, D. (2017). Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_21
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