extreme outer connected monophonic graphs

For a connected graph g of order at least two, a set s of vertices in a graph g is said to be an textit{outer connected monophonic set} if s is a monophonic set of g and either s=v or the subgraph induced by v−s is connected. the minimum cardinality of an outer connected monophonic set of g is the textit{outer connected monophonic number} of g and is denoted by moc(g). the number of extreme vertices in g is its textit{extreme order} ex(g). a graph g is said to be an textit{extreme outer connected monophonic graph} if moc(g) = ex(g). extreme outer connected monophonic graphs of order p with outer connected monophonic number p and extreme outer connected monophonic graphs of order p with outer connected monophonic number p−1 are characterized. it is shown that for every pair a,b of integers with 0≤a≤b and b≥2, there exists a connected graph g with ex(g)=a and moc(g)=b. also, it is shown that for positive integers r,d and k≥2 with r