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) 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 FG, 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) for some perfect [1, 2]-factor FG. This class contains all well-covered graphs G without isolated vertices of order n with (n− 1)/2, and in particular all very well-covered graphs. © 2005 Elsevier B.V. All rights reserved.