Upper Edge Detour Monophonic Number of a Graph

Authors

  • P. Titus University College of Engineering Nagercoil.
  • K. Ganesamoorthy University V. O. C. College of Engineering Tuticorin.

DOI:

https://doi.org/10.4067/S0716-09172014000200004

Keywords:

Edge detour monophonic set, edge detour monophonic number, minimal edge detour monophonic set, upper edge detour monophonic set, upper edge detour monophonic number, conjunto de borde de desvío monofónico, número de desvío monofónico de borde.

Abstract

For a connected graph G of order at least two, a path P is called a monophonic path if it is a chordless path. A longest x — y monophonic path is called an x — y detour monophonic path. A set S of vertices of G is an edge detour monophonic set of G if 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).An edge detour monophonic set S ofG is called a minimal edge detour mono-phonic set ifno proper subset ofS is an edge detour monophonic set of G. The upper edge detour monophonic number of G, denoted by edm+(G),is defined as the maximum cardinality of a minimal edge detour monophonic set ofG. We determine bounds for it and characterize graphs which realize these bounds. For any three positive integers b, c and n with 2 ≤ b ≤ n ≤ c, there is a connected graph G with edm(G) = b, edm+(G) = c and a minimal edge detour monophonic set of cardinality n.

Author Biographies

P. Titus, University College of Engineering Nagercoil.

Department of Mathematics.

K. Ganesamoorthy, University V. O. C. College of Engineering Tuticorin.

Department of Mathematics.

References

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

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

[3] A. P. Santhakumaran, P. Titus, K. Ganesamoorthy and P. Balakrishnan, Edge Detour Monophonic Number of a Graph, Proyecciones Journal of Mathematics, Vol. 32, No. 2, pp. 183-198, (2013).

[4] P. Titus, K. Ganesamoorthy and P. Balakrishnan, The Detour Monophonic Number of a Graph, J. Combin. Math. Combin. Comput. 83, pp. 179-188, (2013).

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

Published

2017-03-23

How to Cite

[1]
P. Titus and K. Ganesamoorthy, “Upper Edge Detour Monophonic Number of a Graph”, Proyecciones (Antofagasta, On line), vol. 33, no. 2, pp. 175-187, Mar. 2017.

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