The Connected and Forcing Connected Restrained Monophonic Numbers of a Graph

For a connected graph G = (V,E) of order at least two, a connected restrained monophonic set S of G is a restrained monophonic set such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by mcr(G). We determine bounds for it and find the same for some special classes of graphs. It is shown that, if a, b and p are positive integers such that 3 ≤ a ≤ b ≤ p, p−1 6= a, p−1 6= b, then there exists a connected graph G of order p, mr(G) = a and mcr(G) = b. Also, another parameter forcing connected restrained monophonic number fcrm(G) of a graph G is introduced and several interesting results and realization theorems are proved.