Trees with the second largest number of maximal independent sets

Jou, Min-Jen; Lin, Jenq-Jong

doi:10.1016/j.disc.2009.02.007

Cited by 10 publications

(2 citation statements)

References 5 publications

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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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Section: Introductionmentioning

confidence: 99%

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

Sun,

2024

Mediterr. J. Math.

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“…It was then studied for various families of graphs, including trees, forests, (connected) graphs with at most one cycle, (connected) triangle-free graphs, (k-)connected graphs, bipartite graphs; for a survey see [2]. Jin and Li [3] investigated the second largest number of ( ) mi G among all graphs of order n; Jou and Lin [4] further explored the same problem for trees and forests; Jin and Yan [5] solved the third largest number of ( ) mi G among all trees of order n. A connected graph (respectively, graph) G with vertex set ( ) V G is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex ( )…”

Section: Introduction and Preliminarymentioning

confidence: 99%

The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

Lin¹,

Jou²

2017

OJDM

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A maximal independent set is an independent set that is not a proper subset of any other independent set. A connected graph (respectively, graph) G with vertex set () V G is called a quasi-tree graph (respectively, quasi-forest graph), if there exists a vertex () x V G ∈ such that G x − is a tree (respectively, forest). In this paper, we survey on the large numbers of maximal independent sets among all trees, forests, quasi-trees and quasi-forests. In addition, we further look into the problem of determining the third largest number of maximal independent sets among all quasi-trees and quasi-forests. Extremal graphs achieving these values are also given.

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The second largest number of maximal independent sets in connected graphs with at most one cycle

Jou

2011

J Comb Optim

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Trees with the second largest number of maximal independent sets

Cited by 10 publications

References 5 publications

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

On the Maximum Number of Maximum Independent Sets of Bipartite Graphs

The Number of Maximal Independent Sets in Quasi-Tree Graphs and Quasi-Forest Graphs

The second largest number of maximal independent sets in connected graphs with at most one cycle

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