Journal ArticleDOI
A finiteness theorem for maximal independent sets
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The setS(k) of all graphsG withmi(G) = k and without isolated vertices (exceptG ≅ K1) or duplicated vertices is studied and it is proved that |V(G)| ≤ 2k−1 +k − 2 for anyG inS(K) andk ≥ 2; consequently,S( k) is finite for anyk.Abstract:
Denote bymi(G) the number of maximal independent sets ofG. This paper studies the setS(k) of all graphsG withmi(G) = k and without isolated vertices (exceptG ? K 1) or duplicated vertices. We determineS(1), S(2), andS(3) and prove that |V(G)| ≤ 2 k?1 +k ? 2 for anyG inS(k) andk ? 2; consequently,S(k) is finite for anyk.read more
Citations
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Journal ArticleDOI
The number of maximum independent sets in graphs
Min-Jen Jou,Gerard J. Chang +1 more
TL;DR: In this article, the problem of determining the largest number of maximum independent sets of a graph of order n is studied, and solutions to this problem are given for various classes of graphs, including general graphs, trees, forests, (connected) graphs with at most one cycle, connected graphs and triangle-free graphs.
Journal ArticleDOI
Bounds on the number of vertex independent sets in a graph
TL;DR: In this article, the authors considered the problem of determining the number of vertex independent sets, and showed that the problem is NP-hard and presented several upper and lower bounds in terms of order, size or independence number.
Journal ArticleDOI
The number of maximal independent sets in connected triangle-free graphs
Gerard J. Chang,Min-Jen Jou +1 more
TL;DR: In this article, it was shown that every connected triangle, free graph of order n ⩾ 22 has at most 5 · 2 (n−6)/2 maximal independent sets if n is even (respectively, odd).
Journal ArticleDOI
Note: Graphs with the second largest number of maximal independent sets
Zemin Jin,Xueliang Li +1 more
TL;DR: This paper determines the second largest number of maximal independent sets among all trees and forests of order n>=4 and characterize those extremal graphs achieving these values.
Journal ArticleDOI
On the number of maximum independent sets of graphs
TL;DR: In this article, the authors characterize graphs with n vertices and with maximum number of maximum independent sets provided that α(G) ≤ 2 or α (G) ≥ n-3.
References
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Journal ArticleDOI
On cliques in graphs
J. W. Moon,L. Moser +1 more
TL;DR: In this article, the maximum number of cliques possible in a graph with n nodes is determined and bounds are obtained for the number of different sizes of clique possible in such a graph.
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The number of maximal independent sets in triangle-free graphs
Mihály Hjuter,Zsolt Tuza +1 more
TL;DR: It is proved that every triangle-free graph on n \geq 4 vertices has at most $2 n /2 $ or $5 \cdot 2^{( n - 5 )/2} $ independent sets maximal under inclusion, whether n is even or odd.
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The number of maximal independent sets in a tree
TL;DR: In this paper, the maximal independent sets of vertices that any tree of n vertices can have were shown to have maximal number of maximal independent vertices, where vertices are independent sets.
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The number of maximal independent sets in connected graphs
TL;DR: A theorem of Moon and Moser is generalized to determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, e.g., n > 50.
Journal ArticleDOI
The number of maximal independent sets in a connected graph
TL;DR: The maximum number of maximal independent sets which a connected graph on n vertices can have is determined, and the extremal graphs are completely characterize, thereby answering a question of Wilf.