Which graphs occur as $γ$-graphs?

Abstract

The γ-graph of a graph G is the graph whose vertices are labelled by the minimum dominating sets of G, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size γ(G)) intersect in a set of size γ(G) − 1. We extend the notion of a γ-graph from distance-1-domination to distance-d-domination, and ask which graphs H occur as γgraphs for a given value of d ≥ 1. We show that, for all d, the answer depends only on whether the vertices of H admit a labelling consistent with the adjacency condition for a conventional γ-graph. This result relies on an explicit construction for a graph having an arbitrary prescribed set of minimum distance-d-dominating sets. We then completely determine the graphs that admit such a labelling among the wheel graphs, the fan graphs, and the graphs on at most six vertices. We connect the question of whether a graph admits such a labelling with previous work on induced subgraphs of Johnson graphs. IntroductionIn this paper we consider only finite, loop-free, undirected graphs G without multiple edges. Our main object of study is the γ d -graph of a graph G, which we introduce via the following three definitions.Definition 1.1. Let G be a graph, and let S and T be subsets of the vertex set V (G) of G. The set S distance-d-dominates T if every vertex of T is within distance d in G of some vertex in S. In the case T = V (G), the subset S is a distance-d-dominating set of G. Definition 1.2. A minimum distance-d-dominating set of a graph G is a distance-d-dominating set of smallest size, and this size is the distance-d-domination number γ d (G) of G.These definitions reduce to well-studied domination notions when d = 1: a distance-1-dominating set is a dominating set; a minimum distance-1-dominating set is a minimum dominating set; and the distance-1-domination number γ 1 (G) is the domination number γ(G). The study of domination in graphs spans more than fifty years, with early interpretations that include the number of queens