γ-Graphs of Trees

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.