Volume 3 | Issue 1 | Year 2012 | Article Id. IJMTT-V3I1P505 | DOI : https://doi.org/10.14445/22315373/IJMTT-V3I1P505

A connected monophonic set in a connected graph = (,) is called a minimal connected monophonic set if no proper subset of M is a connected monophonic set of . The upper connected monophonic number mc + (G) is the maximum cardinality of a minimal connected monophonic set of G. Connected graphs of order p with upper connected monophonic number 2 and p are characterized. It is shown that for any positive integers 2 ≤ a < b ≤ c, there exists a connected graph G with m(G) =a, mc (G) = b and mc + (G) = c, where m(G) is the monophonic number and mc(G) is the connected monophonic number of a graph G. Let M be a minimum connected monophonic set of G. A subset T ⊆ M is called a forcing subset for M if M is the unique minimum connected monophonic set containing T. A forcing subset for M of minimum cardinality is a minimum forcing subset of M. The forcing connected monophonic number of M, denoted by fmc(M), is the cardinality of a minimum forcing subset of M. The forcing connected monophonic number of G, denoted by fmc(G), is fmc(G) = min{fmc(M)}, where the minimum is taken over all minimum connected monophonic set M in G. It is shown that for every integers a and b with a < b, and − − > 0, there exists a connected graph G such that, fmc(G) = a and mc (G) = b.

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J. John, P. Arul Paul Sudhahar, "The Upper Connected Monophonic Number and Forcing Connected Monophonic Number of a Graph," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 3, no. 1, pp. 29-33, 2012. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V3I1P505