The upper complement connected monophonic number of a graph

Abstract

For a connected graph G=(V,E) a monophonic set M of G of is said to be a complement connected monophonic set M = V the subgraph (V - M) is connected. The minimum cardinality of a complement connected monophonic set of if M=V or the complement connected monophonic number of G and is denoted by mee(G).A complement connected monophonic set of M in a connected graph G is called a minimal complement connected monophonic set if no proper subset of M is a complement connected monophonic set of G. The upper complement connected monophonic number mee+(G) of G is the maximum cardinality of a minimal complement connected monophonic set of G Some general properties under this concept are studied. The upper complement connected monophonic number of some standard graphs are determined. Some of its general properties are studied. It is shown that for any positive integers 2 ≤ a ≤b, there exists a connected graph G such that mcc(G) = a and mee+(G) =b.