Journal ArticleDOI
The number of maximal independent sets in triangle-free graphs
Mihály Hjuter,Zsolt Tuza +1 more
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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.Abstract:
In this paper, 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 In each case, the extremal graph is unique If the graph is a forest of odd order, then the upper bound can be improved to $2^{( n - 1 )/2} $read more
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Journal ArticleDOI
Graph colorings with local constraints - a survey
TL;DR: This work surveys the literature on those variants of the chromatic number problem where not only a proper coloring has to be found but some further local restrictions are imposed on the color assignment.
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Enumerating maximal independent sets with applications to graph colouring
TL;DR: Improved algorithms for graph colouring and computing the chromatic number of a graph are constructed by giving tight upper bounds on the number of maximal independent sets of size k in graphs with n vertices.
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Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings
Andreas Björklund,Thore Husfeldt +1 more
TL;DR: This work shows that the Exact Satisfiability problem of size l with m clauses can be solved in time 2mlO(1) and polynomial space, and shows how to count the number of perfect matchings in time O(1.732n) and exponential space.
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Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques
TL;DR: The algorithms that are presented are the best known and are a substantial improvement over previous best results.
Journal ArticleDOI
On Independent Sets and Bicliques in Graphs
TL;DR: It is shown that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3), and an exact exponential-time algorithm is used that computes the number of distinct maximal independent sets in a graphs in time O(1.3642n), where n is thenumber of vertices of the input graph.