A short note on the existence of infinite sequences of γ-graphs of graphs

Abstract For a graph G = ( V , E ) , the γ -graph of G , G ( γ ) = ( V ( γ ) , E ( γ ) ) , is the graph whose vertex set is the collection of minimum dominating sets, or γ -sets of G , and two γ -sets are adjacent in G ( γ ) if they differ by a single vertex and the two different vertices are adjacent in G . We consider sequences of γ -graphs, formed by repeatedly taking the γ -graph of a graph. By considering the γ -graphs of powers of complete graphs, we demonstrate an example claimed to be an infinite sequence of γ -graphs is incorrect, and hence open the question of the existence of such a sequence.