The upper connected outer connected monophonic number of a graph

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 cmco+(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 cmco(G)=a and cmco+(G)=b. Also, for any three integers a,b and n with 4≤a≤n≤b, there is a connected graph G with cmco(G)=a and cmco+(G)=b and a minimal connected outer connected monophonic set of cardinality n, where cmco(G) is the connected outer connected monophonic number of a graph.