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## Proyecciones (Antofagasta)

*Print version* ISSN 0716-0917

#### Abstract

SANTHAKUMARAN, A. P.; TITUS, P. and GANESAMOORTHY, K.. Minimal connected restrained monophonic sets in graphs.* Proyecciones (Antofagasta)* [online]. 2022, vol.41, n.4, pp.879-890.
ISSN 0716-0917. http://dx.doi.org/10.22199/issn.0717-6279-4475.

For a connected graph G = (V, E) of order at least two, a connected restrained monophonic set S of G is a restrained monophonic set such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by mcr(G). A connected restrained monophonic set S of G is called a minimal connected restrained monophonic set if no proper subset of S is a connected restrained monophonic set of G. The upper connected restrained monophonic number of G, denoted by m+ cr(G), is defined as the maximum cardinality of a minimal connected restrained monophonic set of G. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for positive integers a, b such that 4 ≤ a ≤ b, there exists a connected graph G such that mcr(G) = a and m+ cr(G) = b.

**Keywords
:
**restrained monophonic set; restrained monophonic number; connected restrained monophonic set; connected restrained monophonic number; minimal connected restrained monophonic set.