Abstract
The set of vertices of a graph is called distance- \( k \) independent if the distance between any two of its vertices is greater than some integer \( k \geq 1 \). In this paper, we describe \( n \)-vertex trees with a given diameter \( d \) that have the maximum and minimum possible number of distance- \( k \) independent sets among all such trees. The maximum problem is solvable for the case of \( 1 < k < d \leq 5 \). The minimum problem is much simpler and can be solved for all \( 1 < k < d < n \).
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REFERENCES
A. S. Pedersen and P. D. Vestergaard, “An upper bound on the number of independent sets in a tree,” Ars Combinatoria 84, 85–96 (2007).
A. Frendrup, A. S. Pedersen, A. A. Sapozhenko, and P. D. Vestergaard, “Merrifield-Simmons index and minimum number of independent sets in short trees,” Ars Combinatoria 111, 85–95 (2013).
A. B. Dainiak, “On the number of independent sets in the trees of a fixed diameter,” Diskretn. Anal. Issled. Oper. 16 (2), 61–73 (2009) [J. Appl. Ind. Math. 4, 163–171 (2010)].
D. S. Taletskii, “Trees of diameter \( 6 \) and \( 7 \) with minimum number of independent sets,” Mat. Zametki 109 (2), 276–289 (2021) [Math. Notes. 109, 280–291 (2021)].
G. Atkinson and A. Frieze, “On the \( b \)-independence number of sparse random graphs,” Comb. Probab. Comput. 13 (3), 295–309 (2004).
A. Abiad, S. M. Cioabă, and M. Tait, “Spectral bounds for the \( k \)-independence number of a graph,” Linear Algebra Appl. 510, 160–170 (2016).
A. Bouchou and M. Blidia, “On the \( k \)-independence number in graphs,” Australas. J. Comb. 59 (2), 311–322 (2014).
S. O, Y. Shi, and Z. Taoqiu, “Sharp upper bounds on the \( k \)-independence number in graphs with given minimum and maximum degree,” Graphs Comb. 37, 393–408 (2020).
Z. Li and B. Wu, “The \( k \)-independence number of \( t \)-connected graphs,” Appl. Math. Comput. 409, 126412 (2021).
M.-J. Jou and J.-J. Lin, “Characterization of the distance- \( k \) independent dominating sets of the \( n \)-path„” Int. J. Contemp. Math. Sci. 13 (6), 231–238 (2018).
Funding
This work was financially supported by the Russian Science Foundation, project 21-11-00194, https://rscf.ru/en/project/21-11-00194/.
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Translated by V. Potapchouck
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Taletskii, D.S. On Trees with a Given Diameter and the Extremal Number of Distance-\(k\) Independent Sets. J. Appl. Ind. Math. 17, 664–677 (2023). https://doi.org/10.1134/S1990478923030195
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DOI: https://doi.org/10.1134/S1990478923030195