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Independent sets in graphs without subtrees with many leaves

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Abstract

A subtree of a graph is called inscribed if no three vertices of the subtree generate a triangle in the graph. We prove that, for fixed k, the independent set problem is solvable in polynomial time for each of the following classes of graphs: (1) graphs without subtrees with k leaves, (2) subcubic graphs without inscribed subtrees with k leaves, and (3) graphs with degree not exceeding k and lacking induced subtrees with four leaves.

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References

  1. V. E. Alekseev, “A Polynomial Algorithm for Finding the Largest Independent Sets in Claw-Free Graphs,” Diskret. Anal. Issled. Oper. Ser. 1, 6 (4), 3–19 (1999).

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  3. V. E. Alekseev and D. S. Malyshev, “Planar Graph Classes with the Independent Set Problem Solvable in Polynomial Time,” Diskret. Anal. Issled. Oper. 15 (1), 3–10 (2008) [J. Appl. Indust. Math. 3 (1), 1–4 2009].

    MathSciNet  MATH  Google Scholar 

  4. V. E. Alekseev and V. A. Talanov, Graphs and Algorithms. Data Structures. Computation Models (Internet-Univ. Inform. Tekhnol., BINOM. Lab. Znanii,Moscow, 2006) [in Russian].

    Google Scholar 

  5. V. E. Alekseev and D. V. Zakharova, “Independent Sets in the Graphs with Bounded Minors of the Extended Incidence Matrix,” Diskret. Anal. Issled. Oper. 17 (1), 3–10 (2010) [J. Appl. Indust. Math. 5 (1), 14–18 2011].

    MathSciNet  MATH  Google Scholar 

  6. A. V. Bankevich and D. V. Karpov, “Bounds of the Number of Leaves of Spanning Trees,” Zap. Nauchn. Sem. St.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 391, 18–34 (2011) [J. Math. Sci., New York 184 (5), 564–572 2012].

    MathSciNet  MATH  Google Scholar 

  7. P. Erdős and G. Szekeres, “A Combinatorial Problem in Geometry,” CompositioMath. 2, 463–470 (1935).

    MathSciNet  MATH  Google Scholar 

  8. D. S. Malyshev, “Classes ofSubcubicPlanarGraphs forWhich the Independent Set Problem Is Polynomially Solvable,” Diskret. Anal. Issled. Oper. 20 (3), 26–44 (2013) [J. Appl. Indust. Math. 7 (4), 537–548 2013].

    Google Scholar 

  9. G. J. Minty, “On Maximal Independent Sets of Vertices in Claw-Free Graphs,” J. Combin. Theory Ser. B 28, 284–304 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  10. N. Sbihi, “Algorithme de recherche d’un stable de cardinalitémaximum dans un graphe sans étoile,” Discrete Math. 29 (1), 53–76 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. N. Shevchenko, Qualitative Problems in Integer Programming (Nauka, Moscow, 1995) [in Russian].

    MATH  Google Scholar 

  12. D. V. Zakharova, “Weighted Independent Sets in Graphs with Bounded Minors of the Extended Incidence Matrix,” in Proceedings of the X InternationalWorkshop “Discrete Mathematics and Its Applications”, January 1–6, 2010, Moscow, Russia (Mekh.-Mat. Fak., Moskov. Gos. Univ., Moscow, 2010), pp. 303–305.

    Google Scholar 

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Authors and Affiliations

  1. Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, korp. 2, Nizhny Novgorod, 603950, Russia

    V. E. Alekseev & D. V. Zakharova

Authors
  1. V. E. Alekseev
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  2. D. V. Zakharova
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Correspondence to V. E. Alekseev.

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Original Russian Text © V.E. Alekseev, D.V. Zakharova, 2016, published in Diskretnyi Analiz i Issledovanie Operatsii, 2016, Vol. 23, No. 1, pp. 5–14.

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Alekseev, V.E., Zakharova, D.V. Independent sets in graphs without subtrees with many leaves. J. Appl. Ind. Math. 10, 1–6 (2016). https://doi.org/10.1134/S1990478916010014

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  • DOI: https://doi.org/10.1134/S1990478916010014

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