On the structure of dominating graphs (original) (raw)

The k-dominating graph D k (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. A graph is a dominating graph if it is isomorphic to D k (G) for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order n ≥ 2 and with G ∼ = D k (G), then k = 2 and G = K 1,n-1 for some n ≥ 4. It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, C 6 and C 8 are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.