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## Proyecciones (Antofagasta)

*Print version* ISSN 0716-0917

#### Abstract

SANTHAKUMARAN, A. P. and RAGHU, T. Venkata. Upper double monophonic number of a graph.* Proyecciones (Antofagasta)* [online]. 2018, vol.37, n.2, pp.295-304.
ISSN 0716-0917. http://dx.doi.org/10.4067/S0716-09172018000200295.

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.