The upper open monophonic number of a graph

Authors

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

DOI:

https://doi.org/10.4067/S0716-09172014000400003

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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[2] G. Chartrand, E. M. Palmer and P. Zhang, The geodetic number of a graph: A survey, Congr. Numer., 156, pp. 37-58, (2002).

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[5] R. Muntean and P. Zhang, On geodomination in graphs, Congr. Numer., 143, pp. 161-174, (2000).

[6] 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).

[7] 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

[8] 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).

Published

2017-03-23

How to Cite

[1]
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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