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## Print version ISSN 0716-0917

### Proyecciones (Antofagasta) vol.37 no.3 Antofagasta Sept. 2018

#### http://dx.doi.org/10.4067/S0716-09172018000300415

Articles

Edge-to-vertex m-detour monophonic number of a graph

1Hindustan Institute of Technology and Science, Department of Mathematics, Chennai - 603 103, India e-mail : apskumar1953@gmail.com

2Anna University, University College of Engineering Nagercoil, Department of Mathematics, Tirunelveli Region Nagercoil - 629 004, India e-mail: titusvino@yahoo.com

3Coimbatore Institute of Technology, Government Aided Autonomous Institution, Department of Mathematics, Coimbatore - 641 014, India e-mail: kvgm_2005@yahoo.co.in

Abstract

For a connected graph G = (V, E) of order at least three, the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. A u − v path of length dm(u, v) is called a u − v detour monophonic. For subsets A and B of V, the m-monophonic distance Dm(A, B) is defined as Dm(A, B) = max{dm(x, y) : x ∈ A, y ∈ B}. A u − v path of length Dm(A, B) is called a A − B m-detour monophonic path joining the sets A, B ⊆ V, where u ∈ A and v ∈ B. A set S ⊆ E is called an edge-to-vertex m-detour monophonic set of G if every vertex of G is incident with an edge of S or lies on a m-detour monophonic path joining a pair of edges of S. The edge-to-vertex mdetour monophonic number Dmev(G) of G is the minimum order of its edge-to-vertex m-detour monophonic sets and any edge-to-vertex m-detour monophonic set of order Dmev(G) is an edge-to-vertex mdetour monophonic basis of G. Some general properties satisfied by this parameter are studied. The edge-to-vertex m-detour monophonic number of certain classes of graphs are determined. It is shown that for positive integers r, d and k ≥ 4 with r < d, there exists a connected graph G such that radm(G) = r, diamm(G) = d and Dmev(G) = k.

Key Words: monophonic distance; m-detour monophonic path; edge-to-vertex m-detour monophonic set; edge-to-vertex m-detour monophonic basis; edge-to-vertex m-detour monophonic number.

References

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fn1* Research work supported by NBHM Project No. NBHM/R.P.29/2015/Fresh/157.

Received: April 2017; Accepted: April 2018