The connected forcing connected vertex detour number of a graph

For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. 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). For a minimum connected x-detour set Sx of G, a subset T ⊆ Sx is called a connected x-forcing subset for Sx if the induced subgraphG[T ] is connected and Sx is the unique minimum connected x-detour set containing T. A connected x-forcing subset for Sx of minimum cardinality is a minimum connected x-forcing subset of Sx. The connected forcing connected x-detour number of Sx, denoted by cfcdx(Sx), is the cardinality of a minimum connected x-forcing subset for Sx. The connected forcing connected x-detour number of G is cfcdx(G) = min cfcdx(Sx), where the minimum is taken over all minimum connected x-detour sets Sx inG. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers 462 A.P. Santhakumaran and P. Titus of some standard graphs are determined. It is shown that for positive integers a, b, c and d with 2 ≤ a < b ≤ c ≤ d, there exists a connected graph G such that the forcing connected x-detour number is a, connected forcing connected x-detour number is b, connected x-detour number is c and upper connected x-detour number is d, where x is a vertex of G.