The vertex detour hull number of a graph

For vertices x and y in a connected graph G, the detour distance D(x, y) is the length of a longest x− y path in G. An x− y path of length D(x, y) is an x − y detour. The closed detour interval ID[x, y] consists of x, y, and all vertices lying on some x − y detour of G; while for S ⊆ V (G), ID[S] = ⋃ x,y∈S ID[x, y]. A set S of vertices is a detour convex set if ID[S] = S. The detour convex hull [S]D is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among subsets S of V (G) with [S]D = V (G). Let x be any vertex in a connected graph G. For a vertex y in G, denoted by ID[y] , the set of all vertices distinct from x that lie on some x − y detour of G; while for S ⊆ V (G), ID[S] x = ⋃ y∈S ID[y] . For x / ∈ S, S is an x-detour convex set if ID[S] x = S. The x-detour convex hull of S, [S]xD is the smallest x-detour convex set containing S. A set S is an x-detour hull set if [S]xD = V (G) − {x} and the minimum cardinality of x-detour hull sets is the x-detour hull number dhx(G) of G. For x / ∈ S, S is an x-detour set of G if ID[S] x = V (G)−{x} and the minimum cardinality of x-detour sets is the x-detour number dx(G) of G. Certain general properties of the x-detour hull number of a graph are studied. It is shown that for Research supported by DST Project No. SR/S4/MS:319/06 322 A.P. Santhakumaran and S.V. Ullas Chandran each pair of positive integers a, b with 2 ≤ a ≤ b+1, there exist a connected graph G and a vertex x such that dh(G) = a and dhx(G) = b. It is proved that every two integers a and b with 1 ≤ a ≤ b, are realizable as the xdetour hull number and the x-detour number respectively. Also, it is shown that for integers a, b and n with 1 ≤ a ≤ n − b and b ≥ 3, there exist a connected graph G of order n and a vertex x such that dhx(G) = a and the detour eccentricity of x, eD(x) = b. We determine bounds for dhx(G) and characterize graphs G which realize these bounds.