On the independence graph of a graph

Vertices of the independence graph of a graph G represent maximum independent sets of G, two vertices being adjacent whenever the corresponding sets are disjoint. Vizingʹs inequality involving the independence number of the Cartesian product of graphs G and H states that α(G□H)⩽min{α(G)|V(H)|,α(H)|V(G)|}. It has been observed by Hell, Yu and Zhou that the equality is achieved precisely when there is a homomorphism from one factor to the independence graph of the other factor. In this note, we prove that every graph is the independence graph of some graph, and obtain some structural properties of independence graphs that enable us to describe a large class of graphs for which α(G□G)=α(G)|V(G)|.