Maximal independent sets in complementary prism graphs

Let G be a graph and (G) over bar be the complement of G. The complementary prism of G, denoted by G (G) over bar, is the graph formed by the disjoint union of G and (G) over bar by adding the edges of a perfect matching between the corresponding vertices of G and (G) over bar. A graph G is well-covered if every maximal independent set of G has the same cardinality. We prove that if G (G) over bar is well-covered and G is not well covered, G has exactly two consecutive sizes of maximal independent sets. We prove other results concerning well-covered complementary prisms.