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On Trees of Bounded Degree with Maximal Number of Greatest Independent Sets

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Abstract

Given n and d, we describe the structure of trees with the maximal possible number of greatest independent sets in the class of n-vertex trees of vertex degree at most d.We show that the extremal tree is unique for all even n but uniqueness may fail for odd n; moreover, for d = 3 and every odd n ≥ 7, there are exactly ⌈(n − 3)/4⌉ + 1 extremal trees. In the paper, the problem of searching for extremal (n, d)-trees is also considered for the 2-caterpillars; i.e., the trees in which every vertex lies at distance at most 2 from some simple path. Given n and d ∈ {3, 4}, we completely reveal all extremal 2-caterpillars on n vertices each of which has degree at most d.

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References

  1. E. Andriantiana, “Energy, Hosoya Index, and Merrifield–Simmons Index of Trees with Prescribed Degree Sequence,” Discrete Appl.Math. 161 (6), 724–741 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Griggs, C. Grinstead, and D. Guichard, “The Number of Maximal Independent Sets in a Connected Graph,” DiscreteMath. 68 (2–3), 211–220 (1988).

    MathSciNet  MATH  Google Scholar 

  3. C. Heuberger and S. Wagner, “Maximizing the Number of Independent Subsets over Trees with Bounded Degree,” J. Graph Theory 58 (1), 49–68 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Hujter and Z. Tuza, “The Number of Maximal Independent Sets in Triangle-Free Graphs,” SIAM J. DiscreteMath. 6 (2), 284–288 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Jou and G. Chang, “Maximal Independent Sets in Graphs with At Most One Cycle,” Discrete Appl. Math. 79 (1–3), 67–73 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Jou and G. Chang, “The number ofmaximum independent sets in graphs, J. Graph Theory 4 (4), 685–695 (2000).

    MATH  Google Scholar 

  7. J. Liu, “Maximal Independent Sets in Bipartite Graphs,” J. Graph Theory 17 (4), 495–507 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Moon and L. Moser, “On Cliques in Graphs,” Israel J. Math. 3, 23–28 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Wilf, “The Number of Maximal Independent Sets in a Tree,” SIAM J. Algebraic DiscreteMethods 7 (1), 125–130 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Zito, “The Structure and Maximum Number of Maximum Independent Sets in Trees,” J. Graph Theory 15 (2), 207–221, (1991).

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Lobachevsky Nizhny Novgorod State University, ul. Gagarina 23, Nizhny Novgorod, 603950, Russia

    D. S. Taletskii & D. S. Malyshev

  2. National Research University Higher School of Economics, Bolshaya Pechyorskaya ul. 25/12, Nizhny Novgorod, 603155, Russia

    D. S. Malyshev

Authors
  1. D. S. Taletskii
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  2. D. S. Malyshev
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Correspondence to D. S. Taletskii.

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Original Russian Text © D.S. Taletskii, D.S. Malyshev, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 2, pp. 101–123.

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Taletskii, D.S., Malyshev, D.S. On Trees of Bounded Degree with Maximal Number of Greatest Independent Sets. J. Appl. Ind. Math. 12, 369–381 (2018). https://doi.org/10.1134/S1990478918020175

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  • DOI: https://doi.org/10.1134/S1990478918020175

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