For vertices x and y in a connected graph G = (V, E) of order at least two, the detour distance D(x, y) is the length of the longest x - y path in G. An x - y path of length D(x, y) is called an x - y detour. For any vertex x in G, a set S subset of V is an x-detour set of G if each vertex v epsilon V lies on an x - y detour for some element y in S. The minimum cardinality of an x-detour set of G is defined as the x-detour number of G, denoted by d(x)(G). An x-detour set of cardinality d(x)(G) is called a d(x)-set of G : A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cd(x)(G). A connected x-detour set of cardinality cd(x)(G) is called a cd(x)-set of G. A connected x-detour set S-x is called a minimal connected x-detour set if no proper subset of S-x is a connected x-detour set. The upper connected x-detour number, denoted by cd(x)(+)(G) is defined as the maximum cardinality of a minimal connected x-detour set of G. We determine bounds for cd(x)(+)(G) and find the same for some special classes of graphs. For any three integers a, b and c with 2 <= a < b <= c, there is a connected graph G with d(x)(G) = a, cd(x)(G) = b and cd(x)(+)(G) = c for some vertex x in G. It is shown that for positive integers R, D and n >= 3 with R < D <= 2R, there exists a connected graph G with detour radius R, detour diameter D and cd(x)(+)(G) = n for some vertex x in G.
