148,308 research outputs found
On the number of maximal independent sets in a graph
Miller and Muller (1960) and independently Moon and Moser (1965) determined
the maximum number of maximal independent sets in an -vertex graph. We give
a new and simple proof of this result
Maximal independent sets and maximal matchings in series-parallel and related graph classes
We provide combinatorial decompositions as well as asymptotic tight estimates for two maximal parameters: the number and average size of maximal independent sets and maximal matchings in seriesparallel graphs (and related graph classes) with n vertices. In particular, our results extend previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988]. We also show that these two parameters converge to a central limit law.Postprint (author's final draft
Maximal Independent Sets and Maximal Matchings in Series-Parallel and Related Graph Classes
We provide combinatorial decompositions as well as asymptotic tight estimates for two maximal parameters: the number and average size of maximal independent sets and maximal matchings in series-parallel graphs (and related graph classes) with n vertices. In particular, our results extend previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988]. We also show that these two parameters converge to a central limit law
The typical structure of maximal triangle-free graphs
Recently, settling a question of Erd\H{o}s, Balogh and
Pet\v{r}\'{i}\v{c}kov\'{a} showed that there are at most
-vertex maximal triangle-free graphs, matching the previously known lower
bound. Here we characterize the typical structure of maximal triangle-free
graphs. We show that almost every maximal triangle-free graph admits a
vertex partition such that is a perfect matching and is an
independent set.
Our proof uses the Ruzsa-Szemer\'{e}di removal lemma, the
Erd\H{o}s-Simonovits stability theorem, and recent results of
Balogh-Morris-Samotij and Saxton-Thomason on characterization of the structure
of independent sets in hypergraphs. The proof also relies on a new bound on the
number of maximal independent sets in triangle-free graphs with many
vertex-disjoint 's, which is of independent interest.Comment: 17 page
Maximal independent sets and maximal matchings in series-parallel and related graph classes
The goal of this paper is to obtain quantitative results on the number and on the size of maximal independent sets and maximal matchings in several block-stable graph classes that satisfy a proper sub-criticality condition. In particular we cover trees, cacti graphs and seriesparallel graphs. The proof methods are based on a generating function approach and a proper singularity analysis of solutions of implicit systems of functional equations in several variables. As a byproduct, this method extends previous results of Meir and Moon for trees [Meir, Moon: On maximal independent sets of nodes in trees, Journal of Graph Theory 1988].Postprint (author's final draft
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