The upper open monophonic number of a graph
DOI:
https://doi.org/10.4067/S0716-09172014000400003Keywords:
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.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).
[3] G. Chartrand, F. Harary, H. C. Swart and P. Zhang, Geodomination in graphs, Bulletin of the ICA, 31, pp. 51-59, (2001).
[4] F. Harary, Graph Theory, Addison-Wesley, (1969).
[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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