For both connected well-covered graphs and well-covered graphs without isolated vertices, there can be no "essential vertices", vertices which belong to every minimum vertex cover.
Some later authors have replaced the connectivity requirement with the weaker requirement that a well-covered graph must not have any isolated vertices.
Complementing the characterization of well-covered simple polyhedra in three dimensions, researchers have also considered the well-covered simplicial polyhedra, or equivalently the well-covered maximal planar graphs.