On the structure of graphs with bounded asteroidal number

A set A subset of or equal to V of the vertices of a graph G = (V, E) is an asteroidal set if for each vertex a is an element of A, the set A\{a} is contained in one component of G-N[a]. The maximum cardinality of an asteroidal set of G, denoted by an(G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k greater than or equal to 1, an (G) less than or equal to k if and only if an (H) less than or equal to k for, every minimal triangulation H of G. A dominating target is a set D of vertices such that D boolean OR S is a dominating set of G for every set S such that G[D boolean OR S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d(T)(x,y) - d(G)(x,y) less than or equal to 3.\D \ - 1 for every pair x, y of vertices and every dominating target D of G.