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## Proyecciones (Antofagasta)

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*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.41 no.4 Antofagasta Aug. 2022

#### http://dx.doi.org/10.22199/issn.0717-6279-4475

Artículos

Minimal connected restrained monophonic sets in graphs

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

^{2}Department of Mathematics, University College of Engineering Nagercoil, Anna University, Tirunelveli Region, Nagercoil-629 004, India. e-mail: titusvino@yahoo.com

^{3}Department of Mathematics, Coimbatore Institute of Technology, Coimbatore-641 014, India. e-mail: kvgm 2005@yahoo.co.in

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 m_{cr}(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 m_{cr}(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

Acknowledgments

Research work was supported by Project No. NBHM/R.P.29/2015/Fresh/157, National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Government of India

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[14] A. P. Santhakumaran , P. Titus andK. Ganesamoorthy , “The Restrained Monophonic Number of a Graph”, TWMS journal of pure and applied mathematics (Online). Accepted. [ Links ]

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[16] A. P. Santhakumaran , P. Titus, K. Ganesamoorthy , andM. Murugan , “The forcing total monophonic number of a graph”, *Proyecciones (Antofagasta)*, vol. 40, no. 2, pp. 561-571, 2021. doi: 10.22199/issn.0717-6279-2021-02-0031
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[17] A. P. Santhakumaran , T. Venkata Raghu andK. Ganesamoorthy , “Minimal Restrained Monophonic Sets in Graphs”, TWMS journal of pure and applied mathematics (Online) , vol. 11, no. 3, pp. 762-771, 2021 [ Links ]

Received: September 01, 2021; Accepted: February 01, 2022