The forcing connected outer connected monophonic number of a graph.

For a connected graph G = (V, E) of order at least two, a connected outer connected monophonic set S of G is an outer connected monophonic set such that the subgraph induced by S is connected. The minimum cardinality of a connected outer connected monophonic set of G is the connected outer connected monophonic number of G and is denoted by cm_co(G). In this paper, we introduce the concepts of forcing connected outer connected monophonic subset and the forcing connected outer connected monophonic number f_com(G) of a graph G. Certain general properties satisfied by this parameter are studied. It is shown that for every pair a, b of integers with 0 a<b and b>a + 3, there exists a connected graph G such that f_com(G) = a and cm_co(G) = b. [ABSTRACT FROM AUTHOR]