Journal ArticleDOI
Matching-perfect and cover-perfect graphs
TLDR
In this paper, it was shown that a graph G has all matchings of equal size if and only if for every matching setλ in G, G\V(λ) does not contain a maximal open path of odd length greater than one, which is not contained in a cycle.Abstract:
It is shown that a graphG has all matchings of equal size if and only if for every matching setλ inG, G\V(λ) does not contain a maximal open path of odd length greater than one, which is not contained in a cycle. (V(λ) denotes the set of vertices incident with some edge ofλ.) Subsequently edge-coverings of graphs are discussed. A characterization is supplied for graphs all whose minimal covers have equal size.read more
Citations
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Journal ArticleDOI
Well-covered graphs: a survey
TL;DR: A graph G is well-covered if every maximal independent set of points in G is also maximum as discussed by the authors, which is equivalent to the property that the greedy algorithm for constructing a maximal independent subset always results in a maximum independent set.
Journal ArticleDOI
Complexity results for well‐covered graphs
TL;DR: It is shown that well-covered graph recognition is co-NP-complete and that several other problems are NP-complete for well- covered graphs.
Book ChapterDOI
A Note on Well-Covered Graphs
TL;DR: In this paper, it was shown that determining if a graph G is not well-covered is an NP-complete problem, and that determining whether a graph is well covered is NP-hard.
Journal ArticleDOI
Well-covered graphs and extendability
Nathaniel Dean,Jennifer Zito +1 more
TL;DR: It is shown that in order to determine whether a graph is well- covered it is sometimes sufficient to verify that it is k-extendable for small values of k, which leads to efficient algorithms for recognizing well-covered graphs.
Journal ArticleDOI
Efficient recognition of equimatchable graphs
Marc Demange,Tınaz Ekim +1 more
TL;DR: A new characterization of equimatchable graphs that are graphs with all maximal matchings having the same size is given and an O ( n 2 m ) -algorithm for deciding whether a general graph of order n and with m edges is equimATCHable is developed.
References
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Journal ArticleDOI
Matchings in Polytopal Graphs.
TL;DR: A number of new results on matchings are obtained, in particular concerning planar and polytopal graphs.
Journal ArticleDOI
A note on line coverings of graphs
TL;DR: It is shown that a minimal cover of agraph is minimum if and only if it contains a maximum matching of that graph; a maximal matching of a graph is maximum if andonly if it is contained in a minimum cover of thatgraph diminished by the set of its isolated points.