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The Structure of Well-Covered Graphs and the Complexity of Their Recognition Problems

✍ Scribed by David Tankus; Michael Tarsi

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 449 KB
Volume: 69
Category: Article
ISSN: 0095-8956
DOI: 10.1006/jctb.1996.1742

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✦ Synopsis

A graph is well-covered if all its maximal independent sets are of the same cardinality. Deciding whether a given graph is well-covered is known to be NP-hard in general, and solvable in polynomial time, if the input is restricted to certain families of graphs. We present here a simple structural characterization of well-covered graphs and then apply it to the recognition problem. Apparently, polynomial algorithms become easier to design. In particular we present a new polynomial time algorithm for the case where the input graph contains no induced subgraph isomorphic to K 1, 3 . Considering the line-graph of an input graph, this result provides a short and simple alternative to a proof by Lesk, Plummer and Pulleyblank, who showed that equimatchable graphs can be recognized in polynomial time.

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