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

TL;DR

This paper proves that not all bipartite graphs, specifically $K_{2,3}$, can be represented as the $\gamma$-graph of some bipartite graph, answering an open question in the theory of $\gamma$-graphs.

Contribution

The paper demonstrates that the bipartite graph $K_{2,3}$ cannot be realized as a $\gamma$-graph of any bipartite graph, providing a counterexample to an open problem.

Findings

$K_{2,3}$ is not a $\gamma$-graph of any bipartite graph

Answer to an open question in $\gamma$-graph theory

Counterexample for bipartite $\gamma$-graph realizability

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.