Geodesic
The connected forcing connected vertex detour number of a graph
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 461-473

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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 subgraph G[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 cf_cdx(Sₓ), is the cardinality of a minimum connected x-forcing subset for Sₓ. The connected forcing connected x-detour number of G is cf_cdx(G) = mincf_cdx(Sₓ), where the minimum is taken over all minimum connected x-detour sets Sₓ in G. Certain general properties satisfied by connected x-forcing sets are studied. The connected forcing connected vertex detour numbers 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.
Keywords: vertex detour number, connected vertex detour number, upper connected vertex detour number, forcing connected vertex detour number, connected forcing connected vertex detour number 
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Santhakumaran, A.; Titus, P. The connected forcing connected vertex detour number of a graph. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 3, pp. 461-473. http://geodesic.mathdoc.fr/item/DMGT_2011_31_3_a3/