The upper connected vertex detour number of a graph

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 ⊆ V is an x-detour set of G if each vertex ν ∈ 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 dx(G): An x-detour set of cardinality dx(G) is called a dx-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 cdx(G): A connected x-detour set of cardinality cdx(G) is called a cdx-set of G: A connected x-detour set Sxis called a minimal connected x-detour set if no proper subset of Sxis a connected x-detour set. The upper connected x-detour number, denoted by c d x + ( G ) , is defined as the maximum cardinality of a minimal connected x-detour set of G: We determine bounds for c d 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 dx(G) = a, cdx(G) = b and c d 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 c d x + ( G ) = n for some vertex x in G: