$\gamma$-Graphs of Trees

Abstract

For a graph $G = (V, E)$, the $\gamma$-graph of $G$, denoted $G(\gamma) = (V(\gamma), E(\gamma))$, is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent in $G(\gamma)$ if they differ by a single vertex and the two different vertices are adjacent in $G$. In this paper, we consider $\gamma$-graphs of trees. We develop an algorithm for determining the $\gamma$-graph of a tree, characterize which trees are $\gamma$-graphs of trees, and further comment on the structure of $\gamma$-graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.