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

• A. P. Santhakumaran Hindustan Institute of Technology and Science.
• P. Titus Anna University.
• K. Ganesamoorthy Coimbatore Institute of Technology.

### Resumen

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

### Biografía del autor

A. P. Santhakumaran, Hindustan Institute of Technology and Science.
Departamento de Matemáticas.
P. Titus, Anna University.
University College of Engineering Nagercoil. Department of Mathematics.
K. Ganesamoorthy, Coimbatore Institute of Technology.
Government Aided Autonomous Institution. Departamento de Matemáticas.

### Citas

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Publicado
2018-09-24
Cómo citar
Santhakumaran, A., Titus, P., & Ganesamoorthy, K. (2018). Edge-to-vertex m-detour monophonic number of a graph. Proyecciones. Revista De Matemática, 37(3), 415-428. Recuperado a partir de http://www.revistaproyecciones.cl/article/view/3161
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