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New Cases of the Polynomial Solvability of the Independent Set Problem for Graphs with Forbidden Paths

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Abstract

The independent set problem is solvable in polynomial time for the graphs not containing the path P k for any fixed k. If the induced path P k is forbidden then the complexity of this problem is unknown for k > 6. We consider the intermediate cases that the induced path P k and some of its spanning supergraphs are forbidden. We prove the solvability of the independent set problem in polynomial time for the following cases: (1) the supergraphs whose minimal degree is less than k/2 are forbidden; (2) the supergraphs whose complementary graph has more than k/2 edges are forbidden; (3) the supergraphs from which we can obtain P k by means of graph intersection are forbidden.

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References

  1. V. E. Alekseev, “On the Local Restriction Effect on the Complexity of Finding the Graph Independence Number,” in Combinatorial and Algebraic Methods in Applied Mathematics (Izd. Gor’kovskogo Univ., Gor’ky, 1982), pp. 3–13.

    Google Scholar 

  2. V. E. Alekseev, “On the Number of Maximal Independent Sets in the Graphs from Hereditary classes," in Combinatorial and Algebraic Methods in Discrete Optimization (Nizhny Novgorod. Gos. Univ., Nizhny Novgorod, 1991), pp. 5–8.

    Google Scholar 

  3. V. E. Alekseev, “A Polynomial Algorithm for Finding the Largest Independent Sets in Fork-Free Graphs, Diskretn. Anal. Issled. Oper. Ser. 1, 6 (4), 3–19 (1999) [Discrete Appl.Math. 135 (1–3), 3–16 (2004)].

    MathSciNet  MATH  Google Scholar 

  4. V. E. Alekseev and D. V. Korobitsyn, “Complexity of Some Problems for Hereditary Classes of Graphs,” Diskretn.Mat. 4 (4), 34–40 (1992).

    MathSciNet  MATH  Google Scholar 

  5. D. S. Malyshev and D. V. Sirotkin, “Polynomial-TimeSolvability of the Independent Set Problem in a Certain Class of Subcubic PlanarGraphs,” Diskretn.Anal. Issled.Oper. 24 (3), 35–60 (2017) [J. Appl. Indust.Math. 11 (3), 400–414 (2017)].

    MATH  Google Scholar 

  6. A. Grzesik, T. Klimošová, Mar. Pilipczuk, and Mic. Pilipczuk, “Polynomial-Time Algorithm for Maximum Weight Independent Set on P 6-Free Graphs” (Cornell Univ. Libr. e-Print Archive, arXiv:1707.05491, 2017).

    Google Scholar 

  7. W. Hsu, Y. Ikura, and G. L. Nemhauser, “A Polynomial Algorithm forMaximumWeighted Vertex Packing in Graphs without Long Odd Cycles,” Math. Program. 20, 225–232 (1981).

    Article  MATH  Google Scholar 

  8. T. Karthick and F. Maffray, “Weighted Independent Sets in Classes of P 6-FreeGraphs,” Discrete Appl. Math. 209, 217–226 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Lokshtanov, M. Vatshelle, and Y. Villanger, “Independent Set in P 5-Free Graphs in Polynomial Time,” in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms SODA-2014 (Portland, OR, USA, January 5–7, 2014) (SIAM, Philadelphia, PA, 2014), pp. 570–581.

    Google Scholar 

  10. V. V. Lozin, J. Monnot, and B. Ries, “On theMaximum Independent Set Problemin Subclasses of Subcubic Graphs,” J. Discrete Algorithms 31, 104–112 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. V. Lozin and R. Mosca, “Independent Sets in Extension of 2K 2-Free Graphs,” Discrete Appl. Math. 146 (1), 74–80 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  12. V. V. Lozin and D. Rautenbach, “Some Results on Graphs without Long Induced Paths,” Inform. Process. Lett. 88, 167–171 (2003).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Information Technology, Mathematics and Mechanics, Lobachevskii Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia

    V. E. Alekseev & S. V. Sorochan

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  1. V. E. Alekseev
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  2. S. V. Sorochan
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Correspondence to V. E. Alekseev.

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Original Russian Text © V.E. Alekseev, S.V. Sorochan, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 2, pp. 5–18.

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Alekseev, V.E., Sorochan, S.V. New Cases of the Polynomial Solvability of the Independent Set Problem for Graphs with Forbidden Paths. J. Appl. Ind. Math. 12, 213–219 (2018). https://doi.org/10.1134/S1990478918020023

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  • DOI: https://doi.org/10.1134/S1990478918020023

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