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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.40 no.2 Antofagasta abr. 2021 


The forcing total monophonic number of a graph

1Hindustan Institute of Technology and Science. Dept. of Mathematics, Chennai, TN, India. E-mail:

2University College of Engineering Nagercoil, Dept. of Mathematics, Nagercoil, TN, India. E-mail:

3Coimbatore Institute of Technology, Dept. of Mathematics, Coimbatore , TN, India. E-mail:

4Coimbatore Institute of Technology, Dept. of Mathematics, Coimbatore , TN, India. E-mail:


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

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


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

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