The Upper Connected Monophonic Number and Forcing Connected Monophonic Number of a Graph International Journal of Mathematical Trends and Technology (IJMTT) © 2012 by IJMTT Journal Volume-3 Issue-1 Year of Publication : 2012 Authors : J. John, P. Arul Paul Sudhahar J. John, P. Arul Paul Sudhahar "The Upper Connected Monophonic Number and Forcing Connected Monophonic Number of a Graph"International Journal of Mathematical Trends and Technology (IJMTT),V3(1):29-33.June 2012. Published by Seventh Sense Research Group.

Abstract
A connected monophonic set ࡹ in a connected graph ࡳ = (ࢂ,ࡱ) is called a minimal connected monophonic set if no proper subset of M is a connected monophonic set of ࡳ. 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 ࢈ − ૛ࢇ − ૛ > 0, there exists a connected graph G such that, fmc(G) = a and mc (G) = b.

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Keywords
monophonic number, connected monophonic number, upper connected monophonic number, forcing connected monophonic number.