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/

[1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley, Redwood City, CA, 1990).

[2] G. Chartrand, H. Escuadro and P. Zang, Detour Distance in Graphs, J. Combin. Math. Combin. Comput. 53 (2005) 75-94.

[3] G. Chartrand, F. Harary and P. Zang, On the geodetic number of a graph, Networks 39 (2002) 1-6, doi: 10.1002/net.10007.

[4] G. Chartrand, G.L. Johns and P. Zang, The Detour Number of a Graph, Utilitas Mathematica 64 (2003) 97-113.

[5] G. Chartrand, G.L. Johns and P. Zang, On the Detour Number and Geodetic Number of a Graph, Ars Combinatoria 72 (2004) 3-15.

[6] F. Harary, Graph Theory (Addison-Wesley, 1969).

[7] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17 (1993) 87-95, doi: 10.1016/0895-7177(93)90259-2.

[8] T. Mansour and M. Schork, Wiener, hyper-Wiener detour and hyper-detour indices of bridge and chain graphs, J. Math. Chem. 47 (2010) 72-98, doi: 10.1007/s10910-009-9531-7.

[9] A.P. Santhakumaran and P. Titus, Vertex Geodomination in Graphs, Bulletin of Kerala Mathematics Association 2 (2005) 45-57.

[10] A.P. Santhakumaran and P. Titus, On the Vertex Geodomination Number of a Graph, Ars Combinatoria, to appear.

[11] A.P. Santhakumaran and P. Titus, The Vertex Detour Number of a Graph, AKCE International Journal of Graphs and Combinatorics 4 (2007) 99-112.

[12] A.P. Santhakumaran and P. Titus, The Connected Vertex Geodomination Number of a Graph, Journal of Prime Research in Mathematics 5 (2009) 101-114.

[13] A.P. Santhakumaran and P. Titus, The Connected Vertex Detour Number of a Graph, communicated.

[14] A.P. Santhakumaran and P. Titus, The Upper Connected Vertex Detour Number of a Graph, communicated.