On the Structure of Dominating Graphs

The k-dominating graph Dk(G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. In this paper, after presenting several basic properties of k-dominating graphs, it is proved that if G is a graph with no isolates, of order n ≥ 2, and with G ∼= Dk(G), then k = 2 and G = K1,n−1 for some n ≥ 4. The problem of which graphs are dominating graphs is also considered. In particular it is shown that paths and cycles of order at least 5 are not dominating graph. Some results on the order of dominating graphs are also obtained.