The number of maximum independent sets in graphs
Min-Jen Jou,Gerard J. Chang +1 more
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TLDR
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.Abstract:
In this paper, we study the problem of determining the largest number of maximum independent sets of a graph of order n. 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. Extremal graphs achieving the maximum values are also given.read more
Citations
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The number of independent sets in unicyclic graphs
TL;DR: Upper and lower bounds for the number of independent sets in a unicyclic graph in terms of its order are determined and the extremal graphs are characterized.
Journal ArticleDOI
On the Maximum Number of Cliques in a Graph
TL;DR: The maximum number of cliques in a graph for the following graph classes is determined: graphs with n vertices and m edges, d-degenerate graphs, and planar graphs.
Journal ArticleDOI
Extractive multi-document text summarization based on graph independent sets
Taner Uçkan,Ali Karci +1 more
TL;DR: The Maximum Independent Set, which has not been used previously in any summarization study, has been utilized within the context of this study and a text processing tool is suggested in order to preserve the semantic cohesion between sentences in the representation stage of introductory texts.
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 IssueDOI
Maximal and maximum independent sets in graphs with at most r cycles
Bruce E. Sagan,Vincent Vatter +1 more
TL;DR: In this article, the problem of determining the maximum value of m(G) over all connected graphs with n vertices and at most r cycles has been studied, and it has been shown that c(n,r) is the optimal solution when r is large relative to n.
References
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Book
Algorithmic graph theory and perfect graphs
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
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.
Journal ArticleDOI
On generating all maximal independent sets
TL;DR: An algorithm is presented that generates all maximal independent sets of a graph in lexicographic order, with only polynomial delay between the output of two successive independent sets, unless P=NP.
Journal ArticleDOI
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.
Journal ArticleDOI
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.