On graphs with the third largest number of maximal independent sets

Hua, Hongbo; Hou, Yaoping

doi:10.1016/j.ipl.2008.10.013

Cited by 7 publications

(4 citation statements)

References 13 publications

(17 reference statements)

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“…Set i(n) = mi(I (n)) and i (n) = mi(I (n)). By Lemma 1.3, it is easy to obtain that Note that Hua and Hou [6] obtained that i (n) = 97 108 g(n) if n = 3s + 1 and 70 81 g(n) otherwise, which is not correct by direct calculation. It is easy to see Note that I (n) G(n), H (n), hence in view of (1.2), Theorem 3.1 in [6] is not true.…”

Section: Theorem 15 ([45]) If G Is a Graph With N Vertices And G G(mentioning

confidence: 98%

“…By Lemma 1.3, it is easy to obtain that Note that Hua and Hou [6] obtained that i (n) = 97 108 g(n) if n = 3s + 1 and 70 81 g(n) otherwise, which is not correct by direct calculation. It is easy to see Note that I (n) G(n), H (n), hence in view of (1.2), Theorem 3.1 in [6] is not true. The following result characterizes the third largest number of maximal independent sets of n-vertex graphs (n 3), the corresponding extremal graphs are identified.…”

Section: Theorem 15 ([45]) If G Is a Graph With N Vertices And G G(mentioning

confidence: 98%

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On the Third Largest Number of Maximal Independent Sets of Graphs

Zhang

2015

Bull. Malays. Math. Sci. Soc.

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Section: Theorem 15 ([45]) If G Is a Graph With N Vertices And G G(mentioning

confidence: 98%

Section: Theorem 15 ([45]) If G Is a Graph With N Vertices And G G(mentioning

confidence: 98%

On the Third Largest Number of Maximal Independent Sets of Graphs

Zhang

2015

Bull. Malays. Math. Sci. Soc.

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“…Complexity Analysis e externally independent set problem of graphs has been studied widely and many results have been proposed. Most of the existing results can be roughly classified, from viewpoint of research approach, into three categories: graph theory-based, integer programming model-based, and algorithm-based [29].…”

Section: Comparisons With Existing Approaches Andmentioning

confidence: 99%

Matrix Formulation of EISs of Graphs and Its Application to WSN Covering Problems

Yan

Yue

Deng

2021

Journal of Mathematics

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In this paper, the problem of formulating and finding externally independent sets of graphs is considered by using a newly developed STP method, called semitensor product of matrices. By introducing a characteristic value of a vertex subset of a graph and using the algebraic representation of pseudological functions, several necessary and sufficient conditions of matrix form are proposed to express the externally independent sets (EISs), minimum externally independent sets (MEISs), and kernels of graphs. Based on this, the concepts of EIS matrix, MEIS matrix, and kernel matrix are introduced. By these matrices’ complete characterization of these three structures of graphs, three algorithms are further designed which can find all these kinds of subsets of graphs mathematically. The results are finally applied to a WSN covering problem to demonstrate the correctness and effectiveness of the proposed results.

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“…Li, Zhang and Zhang [12] determined bipartite graphs with at least one cycle having the maximal number of maximal independent sets. For more advances on maximal independent sets, we refer the readers to [1,5,9,11] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

Sun,

2024

Mediterr. J. Math.

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An independent set in a graph G is a set of pairwise non-adjacent vertices of G. The independence number, α, of G is the maximum cardinality of an independent set in G. An independent set in G is maximum if it has cardinality α. Mohr and Rautenbach determined the n-vertex trees (resp. disconnected graphs) with independence number α having the largest number of maximum independent sets. Quite recently, Mohr and Rautenbach characterized the connected graphs of order n and independence number α that maximize the number of maximum independent sets. As a continuance of these works, we characterize the bipartite graphs (containing at least one cycle) of order n and independence number α having the maximum number of maximum independent sets. Furthermore, we determine the n-vertex trees with independence number α having the second and third largest number of maximum independent sets. We also characterize the n-vertex forests with independence number α having the first three largest number of maximum independent sets. Among the n-vertex disconnected graphs with independence number α we identify the graphs with the second largest number of maximum independent sets. Our result gives a solution to an open problem of Derikvand and Oboudi [4], and we obtain a complete classification of connected graphs with order n and independence number n − 3.

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On graphs with the third largest number of maximal independent sets

Cited by 7 publications

References 13 publications

On the Third Largest Number of Maximal Independent Sets of Graphs

On the Third Largest Number of Maximal Independent Sets of Graphs

Matrix Formulation of EISs of Graphs and Its Application to WSN Covering Problems

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

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