Abstract
Abstract : Plummer introduced the concept of a well-covered graph in 1970. A graph is well-covered if every maximal independent set (with respect to set inclusion) in the graph is also a maximum independent set. Various subclasses of well-covered graphs have been studied. We consider the subclass which we call strongly well-covered graphs. A strongly well-covered graph G is a well-covered graph with the additional property that G-e is also well-covered for every edge e in G. By making use of (1) structural characteristics of strongly well-covered graphs and (2) the theory of Euler contributions (for planar graphs), we show that there are only four planar strongly well-covered graphs.
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