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## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.32 no.2 Antofagasta May 2013

#### http://dx.doi.org/10.4067/S0716-09172013000200007

Proyecciones Journal of Mathematics Vol. 32, N^{o} 2, pp. 183-198, June 2013. Universidad Católica del Norte Antofagasta - Chile

**Edge Detour Monophonic Number of a Graph**

A. P. Santhakumaran

* Hindustan University, India *

P. Titus K. Ganesamoorthy

P. Balakrishnan

*University College of Engineering Nagercoil, India *

**ABSTRACT**

For a connected graph G of order at least two, an edge detour monophonic set of G is a set S of vertices such that every edge of G lies on a detour monophonic path joining some pair of vertices in S. The edge detour monophonic number of G is the minimum cardinality of its edge detour monophonic sets and is denoted by edm(G) .We determine bounds for it and characterize graphs which realize these bounds. Also, certain general properties satisfied by an edge detour monophonic set are studied. It is shown that for positive integers a, b and c with 2 ≤ a ≤ c, there exists a connected graph G such that m(G) = a, m!(G) = b and edm(G) = c,where m(G) is the monophonic number and m! (G) is the edge monophonic number of G. Also, for any integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that dm(G) = a and edm(G)= b,where dm(G) is the detour monophonic number of a graph G.

**Key Words : ***monophonic number, edge monophonic number, detour monophonic number, edge detour monophonic number.*

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

**REFERENCES**

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[2] G. Chartrand, F. Harary and P. Zhang, *On the geodetic number of a* *graph, *Networks, 39 (1), pp. 1-6, (2002). [ Links ]

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[6] T. Mansour and M. Schork, *Wiener, hyper-Wiener detour and hyper detour indices of bridge and chain graphs, *J. Math. Chem., 47, pp.72-98, (2010). [ Links ]

[7] A.P. Santhakumaran, P. Titus and P. Balakrishnan, Some Realisation Results on Edge Monophonic Number of a Graph, communicated. [ Links ]

[8] A.P. Santhakumaran, P. Titus and K. Ganesamoorthy, On the Mono-phonic Number of a Graph, communicated. [ Links ]

[9] P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Mono-phonic Number of a Graph, J. Combin. Math. Combin. Comput., 84, pp. 179-188, (2013). [ Links ]

[10] P. Titus and K. Ganesamoorthy, On the Detour Monophonic Number of a Graph, Ars Combinatoria, to appear. [ Links ]

**A. P. Santhakumaran**

Department of Mathematics Hindustan University

Hindustan Institute of Technology and Science

Chennai - 603 103, India

e-mail: apskumar1953@yahoo.co.in

**P. Titus**

e-mail : titusvino@yahoo.com

**K. Ganesamoorthy**

e-mail : kvgm_2005@yahoo.co.in

**P. Balakrishnan**

Department of Mathematics

University College of Engineering Nagercoil

Anna University,

Tirunelveli Region

Nagercoil - 629 004,

India

e-mail : gangaibala1@yahoo.com

*Received : December 2012. Accepted : April 2013*