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.

Let G = (V, E) be a simple graph. A set I ⊆ V is an independent set, if no two of its members are adjacent in G. The k-independent graph of G, I k (G), is defined to be the graph whose vertices correspond to the independent sets of G that have cardinality at most k. Two vertices in I k (G) are adjacent if and only if the corresponding independent sets of G differ by either adding or deleting a single vertex. In this paper, we obtain some properties of I k (G) and compute it for some graphs.Mathematics Subject Classification: 05C60, 05C69.

No abstract

Let G = (V, E) be a graph. A set S ⊆ V (G) is a dominating set, if every vertex in V (G)\S is adjacent to at least one vertex in S. The k-dominating graph of G, D k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. In this paper we consider and study the n-dominating graph of specific graphs.

No abstract