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## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.33 no.2 Antofagasta June 2014

#### http://dx.doi.org/10.4067/S0716-09172014000200002

**The forcing connected detour number of a graph**

**A. P. Santhakumaran**

*Hindustan University*

*India*

**S. Athisayanathan **

*St. Xavier's College*

*India *

**ABSTRACT**

*For two vertices u and v in a graph G* = (V, E), *the detour distance D(u,* v) *is the length of a longest u—v path in G.A u—v path of length D(u,* v) *is called a u—v detour. A set ⊆ * V *is called a detour set of G if every vertex in G lies on a detour joining a pair of vertices of S.The detour number dn(G) of G is the minimum order of its detour sets and any detour set of order dn(G) is a detour basis of G.A set ⊆* V *is called a connected detour set of G if S is detour set of G and the subgraph* G[S] *induced by S is connected. The connected detour number cdn(G) of G is the minimum order of its connected detour sets and any connected detour set of order cdn(G) is called a connected detour basis of G.A subset T of a connected detour basis S is called a forcing subset for S if S is theuniquecon-nected detour basis containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing connected detour number of S, denoted by fcdn(S), is the cardinality of a minimum forcing subset for S. The forcing connected detour number of G, denoted by fcdn(G),is fcdn(G) = min{fcdn(S)},where the minimum is taken over all connected detour bases S in G. The forcing connected detour numbers ofcertain standard graphs are obtained. It is shown that for each pair a, b of integers with* 0 ≤ *a<b and b* ≥ 3, *there is a connected graph G with fcdn(G) = a and cdn(G) = b.*

**Key Words :** *Detour, connected detour set, connected detour basis, connected detour number, forcing connected detour number.* **AMS**

**Subject Classification :** *05C12.*

**REFERENCES**

[1] F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Reading MA, (1990).

[2] G. Chartrand, H. Escuadro and P. Zhang, Detour distance in graphs, J. Combin. Math. Combin. Comput., 53, pp. 75-94, (2005).

[3] G. Chartrand, L. Johns, and P. Zhang, Detour Number of a Graph, Util. Math. **64,** pp. 97—113, (2003).

[4] G. Chartrand and P. Zhang, Distance in Graphs—Taking the Long View, AKCE J. Graphs.1. No.1, pp. 1—13, (2004).

[5] G. Chartrand and P. Zang, Introduction to Graph Theory, Tata McGraw-Hill, (2006).

[6] A. P. Santhakumaran and S. Athisayanathan, The connected detour number of a graph, J. Combin. Math. Combin. Comput., 69, pp. 205— 218, (2009).

**A. P. Santhakumaran**

*Department of Mathematics Hindustan University*

*Hindustan Institute of Technology and Science Padur, Chennai-603 103, India e-mail : apskumar1953@yahoo.co.in*

**S. Athisayanathan**

*Department of Mathematics*

*St. Xavier's College (Autonomous)*

*Palayamkottai - 627 002*

*India*

*e-mail: athisayanathan@yahoo.co.in*

Received : August 2013. Accepted : December 2013