Publication

Abstract : 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 I D[x, y] consists of x, y, and all vertices lying on some x-y detour of G; while for S ⊆ V(G), I D[S] = ∪ x,y∈S I D [x, y]. A set S of vertices is a detour convex set if I D [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 I D[y] x, the set of all vertices distinct from x that lie on some x-y detour of G; while for S ⊆ V(G), I D[S] x = ∪ y∈S I D[y] x. For x ∉ S, S is an x-detour convex set if I D[S] x = S. The x-detour convex hull of S, [S] x D is the smallest x-detour convex set containing S. A set S is an x-detour hull set if [S] D x = V(G) - {x} and the minimum cardinality of x-detour hull sets is the x-detour hull number dh x(G) of G. For x ∉ S, S is an x-detour set of G if I D [S] x = V(G) - {x} and the minimum cardinality of x-detour sets is the x-detour number d x(G) of G. Certain general properties of the x-detour hull number of a graph are studied. It is shown that for 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 dh x(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 dh x(G) = a and the detour eccentricity of x, e D(x) = b. We determine bounds for dh x(G) and characterize graphs G which realize these bounds.