# The total detour monophonic number of a graph.

## Authors

• A. P. Santhakumaran Hindustan University.
• P. Titus University College of Engineering Nagercoil.
• K. Ganesamoorthy Anna University.

## Keywords:

Detour monophonic set, Detour monophonic number, Total detour monophonic set, Total detour monophonic number

## Abstract

For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. 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 a detour monophonic set of G if each vertex v of G lies on an x - y detour monophonic path for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A total detour monophonic set of cardinality dmt(G) is called a dmt-set of G. We determine bounds for it and characterize graphs which realize the lower bound. It is shown that for positive integers r, d and k ≥ 6 with r < d there exists a connected graph G with monophonic radius r, monophonic diameter d and dmt(G) = k. For positive integers a, b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dmt(G) = b. Also, if p, d and k are positive integers such that 2 ≤ d ≤ p - 2, 3 ≤ k ≤ p and p – d – k + 3 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and dmt(G) = k.

## Author Biographies

### A. P. Santhakumaran, Hindustan University.

Department of Mathematics.

### P. Titus, University College of Engineering Nagercoil.

Department of Mathematics.

### K. Ganesamoorthy, Anna University.

Department of Mathematics, Coimbatore Institute of Technology. Government Aided Autonomous Institution Coimbatore.

## References

BUCKLEY, F. (1990) Distance in Graphs. Redwood City, CA: Addison-Wesley.

DOURADO, M. C. (2008) Algorithmic Aspects of Monophonic Convexity. EN: Electronic Notes in Discrete Mathematics, 30. [s.l.: s.n.], 177-182.

HARARY, F. (1969) Graph Theory. [s.l.]: Addison-Wesley.

SANTHAKUMARAN, A. P. (2011) Monophonic Distance in Graphs. EN: Discrete Mathematics, Algorithms and Applications, 3(2). [s.l.: s.n.], 159-169.

SANTHAKUMARAN, A. P. (2012) A Note on “Monophonic Distance in Graphs”. EN: Discrete Mathematics, Algorithms and Applications, 4(2). [s.l.: s.n.].

TITUS, P. (2016) On the Detour Monophonic Number of a Graph. EN: Ars Combinatoria, 129. [s.l.: s.n.], 33-42.

TITUS, P. (2013) The Detour Monophonic Number of a Graph. EN: J. Combin. Math. Combin. Comput., 84. [s.l.: s.n.], 179-188.

TITUS, P. (2016) The Connected Detour Monophonic Number of a Graph. EN: TWMS Journal of Applied and Engineering Mathematics, 6(1). [s.l.: s.n.], 75-86.

2017-06-02

## How to Cite

[1]
A. P. Santhakumaran, P. Titus, and K. Ganesamoorthy, “The total detour monophonic number of a graph.”, Proyecciones (Antofagasta, On line), vol. 36, no. 2, pp. 209-224, Jun. 2017.

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