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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.37 no.2 Antofagasta June 2018

http://dx.doi.org/10.4067/S0716-09172018000200295 

Articles

Upper double monophonic number of a graph

A. P. Santhakumaran1 

T. Venkata Raghu2 

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

2Sasi Institute of Technology and Engineering, Department of Applied Sciences and Humanities, Tadepalligudem 534 101, India. e-mail : tvraghu2010@gmail.com

Abstract:

A set S of a connected graph G of order n is called a double monophonic set of G if for every pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u-v monophonic path. The double monophonic number dm(G) of G is the minimum cardinality of a double monophonic set. A double monophonic set S in a connected graph G is called a minimal double monophonic set if no proper subset of S is a double monophonic set of G. The upper double monophonic number of G is the maximum cardinality of a minimal double monophonic set of G, and is denoted by dm+(G). Some general properties satisfied by upper double monophonic sets are discussed. It is proved that for a connected graph G of order n, dm(G)=n if and only if dm+(G)=n. It is also proved that dm(G)=n-1 if and only if dm+(G)=n-1 for a non-complete graph G of order n with a full degree vertex. For any positive integers 2 ≤ a ≤ b, there exists a connected graph G with dm(G)= a and dm+(G)=b.

Keywords:  Double monophonic set; double monophonic number; upper double monophonic set; upper double monophonic number

2010 Mathematics Subject Classification: 05C12.

Texto completo disponible sólo en PDF.

Full text available only in PDF format.

Acknowledgements:

The authors are thankful to the referee for his valuable comments.

References:

[1] F. Buckley and F. Harary, Distance in Graphs, Addison - Wesley, Redwood city, CA, (1990). [ Links ]

[2] G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39, pp. 1-6, (2002). [ Links ]

[3] F. Harary, Graph Theory, Addision-Wesley, U.S.A.,(1969). [ Links ]

[4] F. Harary, E. Loukakis and C. Tsouros, The geodetic number of a graph, Math. Comput. Modeling 17, pp. 89 - 95, (1993). [ Links ]

[5] A. P. Santhakumaran and T. Jebaraj, The upper double geodetic number of a graph, Malaysian Journal of Science 30 (3): 225- 229, (2011). [ Links ]

[6] A. P. Santhakumaran and T. Jebaraj, The double geodetic number of a graph, Discuss. Math. Graph Theory, 32, pp. 109-119, (2012). [ Links ]

[7] A. P. Santhakumaran and T. Venkata Raghu, The double monophonic number of a graph, International Journal of Computational and Applied Mathematics, 11 (1), pp. 21-26, (2016). [ Links ]

Received: September 2017; Accepted: February 2018

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License