Abstract
A graph is well-covered if every maximal independent set is a maximum independent set. A strongly well-covered graph G has the additional property that G- e is also well-covered for every line e in G. Hence, the strongly well-covered graphs are a subclass of the well-covered graphs. We characterize strongly well-covered graphs with independence number two and determine a parity condition for strongly well-covered graphs with independence number three. More generally, we show that a strongly well-covered graph (with more than four points) is 3-connected and has minimum degree at least four.
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