# The upper open monophonic number of a graph

## Authors

• A. P. Santhakumaran Hindustan University.
• M. Mahendran Hindustan University.

## Keywords:

Distance, geodesic, geodetic number, open geodetic number, monophonic number, open monophonic number, upper open monophonic number, distancia, geodesia, número geodésico, número geodésico abierto, número monofónico, número monofónico abierto.

## Abstract

For a connected graph G of order n,a subset S of vertices of G is a monophonic set of G if each vertex v in G lies on a x-y monophonic path for some elements x and y in S. The minimum cardinality of a monophonic set of G is defined as the monophonic number of G, denoted by m(G). A monophonic set of cardinality m(G) is called a m-set of G.A set S of vertices of a connected graph G is an open monophonic set of G if for each vertex v in G ,either v is an extreme vertex of G and v G S,or v is an internal vertex of a x-y mono-phonic path for some x,y G S. An open monophonic set of minimum cardinality is a minimum open monophonic set and this cardinality is the open monophonic number, om(G). An open monophonic set S of vertices in a connected graph G is a minimal open monophonic .set if no proper subset of S is an open monophonic set of G.The upper open monophonic number om+ (G) is the maximum cardinality of a minimal open monophonic set of G. The upper open monophonic numbers of certain standard graphs are determined. It is proved that for a graph G of order n, om(G) = n if and only if om+(G)= n. Graphs G with om(G) = 2 are characterized. If a graph G has a minimal open monophonic set S of cardinality 3, then S is also a minimum open monophonic set of G and om(G) = 3. For any two positive integers a and b with 4 < a < b, there exists a connected graph G with om(G) = a and om+(G) = b.

## Author Biographies

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

Department of Mathematics Hindustan University.Hindustan Institute of Technology and Science Padur..

### M. Mahendran, Hindustan University.

Department of Mathematics Hindustan University.Hindustan Institute of Technology and Science Padur.

## References

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 A. P. Santhakumaran and T. Kumari Latha, On the open geodetic number of a graph, SCIENTIA Series A: Mathematical Sciences, Vol. 20, pp. 131-142, (2010).

 A. P. Santhakumaran and M. Mahendran, The connected open monophonic number of a graph, International Journal of Computer Applications (0975-8887), Vol. 80 No. 1, pp. 39-42, (2013).The Upper Open Monophonic Number of a Graph 403

 A. P. Santhakumaran and M. Mahendran, The open monophonic number of a graph, International Journal of Scientific & Engineering Research, Vol. 5 No. 2, pp. 1644-1649, (2014).

2017-03-23

## How to Cite


A. P. Santhakumaran and M. Mahendran, “The upper open monophonic number of a graph”, Proyecciones (Antofagasta, On line), vol. 33, no. 4, pp. 389-403, Mar. 2017.

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