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## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.40 no.2 Antofagasta Apr. 2021

#### http://dx.doi.org/10.22199/issn.0717-6279-2021-02-0031

Artículos

The forcing total monophonic number of a graph

^{1}Hindustan Institute of Technology and Science. Dept. of Mathematics, Chennai, TN, India. E-mail: apskumar1953@gmail.com

^{2}University College of Engineering Nagercoil, Dept. of Mathematics, Nagercoil, TN, India. E-mail: titusvino@yahoo.com

^{3}Coimbatore Institute of Technology, Dept. of Mathematics, Coimbatore , TN, India. E-mail: kvgm_2005@yahoo.co.in

^{4}Coimbatore Institute of Technology, Dept. of Mathematics, Coimbatore , TN, India. E-mail: jrfmaths@gmail.com

*For a connected graph G = (V, E) of order at least two, a subset T of a minimum total monophonic set S of G is a forcing total monophonic subset for S if S is the unique minimum total monophonic set containing T . A forcing total monophonic subset for S of minimum cardinality is a minimum forcing total monophonic subset of S. The forcing total monophonic number f*
_{
tm
}
*(S) in G is the cardinality of a minimum forcing total monophonic subset of S. The forcing total monophonic number of G is f*
_{
tm
}
*(G) = min{f*
_{
tm
}
*(S)}, where the minimum is taken over all minimum total monophonic sets S in G. We determine bounds for it and find the forcing total monophonic number of certain classes of graphs. It is shown that for every pair a, b of positive integers with 0 ≤ a < b and b ≥ a+4, there exists a connected graph G such that f*
_{
tm
}
*(G) = a and m*
_{
t
}
*(G) = b.*

**Keywords: **Total monophonic set; Total monophonic number; Forcing total monophonic subset; Forcing total monophonic number

Acknowledgements

The third author acknowledges support from Research supported by NBHM Project No. NBHM/R.P.29/2015/Fresh/157.

References

[1] F. Buckley and F. Harary, *Distance in graphs*. Redwood City, CA: Addison-Wesley, 1990.
[ Links ]

[2] F. Harary, *Graph theory*. Reading, MA: Addison-Wesley, 1969. [On line]. Available: https://bit.ly/38wIHRk
[ Links ]

[3] K. Ganesamoorthy, "A study of monophonic number and its variants", Ph.D. thesis, Anna University, Chennai, 2013. [ Links ]

[4] E. M. Paluga and S. R. Canoy, “Monophonic numbers of the join and composition of connected graphs”, *Discrete mathematics*, vol. 307, no. 9-10, pp. 1146-1154, 2007, doi: 10.1016/j.disc.2006.08.002
[ Links ]

[5] A. P. Santhakumaran, P. Titus and K. Ganesamoorthy, “On the monophonic number of a graph”, *Journal of applied mathematics & informatics*, vol. 32, no. 1-2, pp. 255-266, 2014, doi: 10.14317/jami.2014.255
[ Links ]

[6] A. P. Santhakumaran , P. Titus, K. Ganesamoorthy, and M. Murugan, “The Total Monophonic Number of a Graph”, in Proceedings of International Conference on Discrete Mathematics and its Applications to Network Science (ICDMANS 2018), *Under process*.
[ Links ]

Received: April 30, 2019; Accepted: January 31, 2021