The upper connected outer connected monophonic number of a graph. Issue 1 (2nd January 2023) - Ganesamoorthy, K.; Lakshmi Priya, S.
Abstract : For a connected graph G of order at least two, a connected outer connected monophonic set S of G is called a minimal connected outer connected monophonic set if no proper subset of S is a connected outer connected monophonic set of G . The upper connected outer connected monophonic number c m c o + ( G ) of G is the maximum cardinality of a minimal connected outer connected monophonic set of G . We determine bounds for it and find the upper connected outer connected monophonic number of certain classes of graphs. It is shown that for any two integers a, b with 4 ≤ a ≤ b ≤ p − 2, there is a connected graph G of order p with c m c o ( G ) = a and c m c o + ( G ) = b . Also, for any three integers a, b and n with 4 ≤ a ≤ n ≤ b, there is a connected graph G with c m c o ( G ) = a and c m c o + ( G ) = b and a minimal connected outer connected monophonic set of cardinality n, where c m c o ( G ) is the connected outer connected monophonic number of a graph.