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## Proyecciones (Antofagasta)

*Print version* ISSN 0716-0917

#### Abstract

SANTHAKUMARAN, A. P.; TITUS, P. and GANESAMOORTHY, K.. The total detour monophonic number of a graph.* Proyecciones (Antofagasta)* [online]. 2017, vol.36, n.2, pp.209-224.
ISSN 0716-0917. http://dx.doi.org/10.4067/S0716-09172017000200209.

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 dm t (G). A total detour monophonic set of cardinality dm t (G) is called a dm t -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 dm t (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 dm t (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 dm t (G) = k.

**Keywords
:
**Detour monophonic set; detour monophonic number; total detour monophonic set; total detour monophonic number.