The forcing total monophonic number of a graph
DOI:
https://doi.org/10.22199/issn.0717-6279-2021-02-0031Keywords:
Total monophonic set, Total monophonic number, Forcing total monophonic subset, Forcing total monophonic numberAbstract
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 ftm(S) in G is the cardinality of a minimum forcing total monophonic subset of S. The forcing total monophonic number of G is ftm(G) = min{ftm(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 ftm(G) = a and mt(G) = b.
References
F. Buckley and F. Harary, Distance in graphs. Redwood City, CA: Addison-Wesley, 1990.
F. Harary, Graph theory. Reading, MA: Addison-Wesley, 1969. [On line]. Available: https://bit.ly/38wIHRk
K. Ganesamoorthy, "A study of monophonic number and its variants", Ph.D. thesis, Anna University, Chennai, 2013.
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
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
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.
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Copyright (c) 2021 A. P. Santhakumaran, P. Titus, K. Ganesamoorthy, M. Murugan
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