Well-covered graphs and factors

Abstract

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α . Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [ 1 , 2 ] -factor F G , i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α ( G ) = α ( F G ) for some perfect [ 1 , 2 ] -factor F G . This class contains all well-covered graphs G without isolated vertices of order n with α ⩾ ( n - 1 ) / 2 , and in particular all very well-covered graphs.