**2010 Mathematics Subject Classification**: 05C12.

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## Proyecciones (Antofagasta)

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*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

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

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

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

Acknowledgements:

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

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Received: September 2017; Accepted: February 2018