A Bipartite Graph That Is Not the $\gamma$-Graph of a Bipartite Graph

Abstract

For a graph $G = (V, E)$, the $\gamma$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent if they differ by a single vertex and the two different vertices are adjacent in $G$. An open question in $\gamma$-graphs is whether every bipartite graph is the $\gamma$-graph of some bipartite graph. We answer this question in the negative by demonstrating that $K_{2, 3}$ is not the $\gamma$-graph of any bipartite graph.