Well-covered graphs and factors
A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality ? ? . Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 9198] 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] [ 1 , 2 ] -factor FG 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 ?(G)=?(FG) ? ( G ) = ? ( F G ) for some perfect [1,2] [ 1 , 2 ] -factor FG F G . This class contains all well-covered graphs G without isolated vertices of order n  with ??(n-1)/2 ? ? ( n - 1 ) / 2 , and in particular all very well-covered graphs.
well-covered factor independence number