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On Subgraph Complementation to H-free Graphs

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Graph-Theoretic Concepts in Computer Science (WG 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12911))

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Abstract

For a class \(\mathcal {G}\) of graphs, the problem Subgraph Complement to \(\mathcal {G}\) asks whether one can find a subset S of vertices of the input graph G such that complementing the subgraph induced by S in G results in a graph in \(\mathcal {G}\). We investigate the complexity of the problem when \(\mathcal {G}\) is H-free for H being a complete graph, a star, a path, or a cycle. We obtain the following results:

  • When H is a \(K_t\) (a complete graph on t vertices) for any fixed \(t\ge 1\), the problem is solvable in polynomial-time. This applies even when \(\mathcal {G}\) is a subclass of \(K_t\)-free graphs recognizable in polynomial-time, for example, the class of \((t-2)\)-degenerate graphs.

  • When H is a \(K_{1,t}\) (a star graph on \(t+1\) vertices), we obtain that the problem is NP-complete for every \(t\ge 5\). This, along with known results, leaves only two unresolved cases - \(K_{1,3}\) and \(K_{1,4}\).

  • When H is a \(P_t\) (a path on t vertices), we obtain that the problem is NP-complete for every \(t\ge 7\), leaving behind only two unresolved cases - \(P_5\) and \(P_6\).

  • When H is a \(C_t\) (a cycle on t vertices), we obtain that the problem is NP-complete for every \(t\ge 8\), leaving behind four unresolved cases - \(C_4, C_5, C_6,\) and \(C_7\).

Further, we prove that these hard problems do not admit subexponential-time algorithms (algorithms running in time \(2^{o(|V(G)|)}\)), assuming the Exponential Time Hypothesis. A simple complementation argument implies that results for \(\mathcal {G}\) are applicable for \(\overline{\mathcal {G}}\), thereby obtaining similar results for H being the complement of a complete graph, a star, a path, or a cycle. Our results generalize two main results and resolve one open question by Fomin et al. (Algorithmica, 2020).

Partially supported by SERB Grant SRG/2019/002276: “Complexity Dichotomies for Graph Modification Problems”.

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References

  1. Aravind, N.R., Sandeep, R.B., Sivadasan, N.: Dichotomy results on the hardness of H-free edge modification problems. SIAM J. Discre. Math. 31(1), 542–561 (2017)

    Google Scholar 

  2. Cai, L., Cai, Y.: Incompressibility of \(H\)-free edge modification problems. Algorithmica 71(3), 731–757 (2014). https://doi.org/10.1007/s00453-014-9937-x

  3. Cai, Y.: Polynomial kernelisation of H-free edge modification problems. M .Phil. thesis, Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong SAR, China (2012)

    Google Scholar 

  4. Cao, Y., Chen, J.: Cluster editing: Kernelization based on edge cuts. Algorithmica 64(1), 152–169 (2012)

    Google Scholar 

  5. Cao, Y., Ke, Y., Yuan, H.: Polynomial Kernels for paw-free edge modification problems. In: Chen, J., Feng, Q., Xu, J. (eds.) TAMC 2020. LNCS, vol. 12337, pp. 37–49. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-59267-7_4

    Chapter  Google Scholar 

  6. Cygan, M., et al.: Parameterized Algorithms. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21275-3

  7. Eiben, E., Lochet, W., Saurabh, S.: A polynomial kernel for paw-free editing. In: Cao, Y., Pilipczuk, M. (eds.) 15th International Symposium on Parameterized and Exact Computation (IPEC 2020), December 14–18, 2020, Hong Kong, China (Virtual Conference). LIPIcs, vol. 180, pp. 10:1–10:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.IPEC.2020.10

  8. Fomin, F.V., Golovach, P.A., Strømme, T.J.F., Thilikos, D.M.: Subgraph complementation. Algorithmica 82(7), 1859–1880 (2020)

    Google Scholar 

  9. Guillemot, S., Havet, F., Paul, C., Perez, A.: On the (non-)existence of polynomial kernels for \(P_\ell \)-free edge modification problems. Algorithmica 65(4), 900–926 (2013)

    Google Scholar 

  10. Guo, J.: A more effective linear kernelization for cluster editing. Theor. Comput. Sci. 410(8–10), 718–726 (2009)

    Google Scholar 

  11. Gyárfás, A.: Generalized split graphs and ramsey numbers. J. Comb. Theory, Ser. A 81(2), 255–261 (1998). https://doi.org/10.1006/jcta.1997.2833

  12. Jelínková, E., Kratochvíl, J.: On switching to \(h\)-free graphs. J. Graph. Theory 75(4), 387–405 (2014)

    Google Scholar 

  13. Kaminski, M., Lozin, V.V., Milanic, M.: Recent developments on graphs of bounded clique-width. Discret. Appl. Math. 157(12), 2747–2761 (2009)

    Google Scholar 

  14. Kolay, S., Panolan, F.: Parameterized algorithms for deletion to \((r, \ell )\)-graphs. In: Harsha, P., Ramalingam, G. (eds.) 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2015). LIPIcs, vol. 45, pp. 420–433. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015). https://doi.org/10.4230/LIPIcs.FSTTCS.2015.420

  15. Kolay, S., Panolan, F.: Parameterized algorithms for deletion to (r, l)-graphs. arXiv preprint arXiv:1504.08120 (2015)

  16. Marx, D., Sandeep, R.B.: Incompressibility of H-free edge modification problems: towards a dichotomy. In: 28th Annual European Symposium on Algorithms (ESA 2020), LIPIcs, vol. 173, pp. 72:1–72:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.ESA.2020.72

  17. Yannakakis, M.: Edge-deletion problems. SIAM J. Comput. 10(2), 297–309 (1981)

    Google Scholar 

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Correspondence to Dhanyamol Antony or Sagartanu Pal .

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Antony, D., Garchar, J., Pal, S., Sandeep, R.B., Sen, S., Subashini, R. (2021). On Subgraph Complementation to H-free Graphs. In: Kowalik, Ł., Pilipczuk, M., Rzążewski, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2021. Lecture Notes in Computer Science(), vol 12911. Springer, Cham. https://doi.org/10.1007/978-3-030-86838-3_9

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  • DOI: https://doi.org/10.1007/978-3-030-86838-3_9

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