The Forcing Circular Number of a Graph

Let S be a cr-set of graph G and let G be a connected graph. If S is the only cr-set that contains T, then a subset T⊆S is referred to be a forcing subset for S. A minimum forcing subset of S is a forcing subset for S of minimum cardinality. The cardinality of a minimum forcing subset of S is the forcing circular number of S, represented by the notation f_cr(S). f_cr (G) = min {f_cr(S)} is the forcing circular number of G, where the minimum is the sum of all minimum forcing circular-sets S in G. For several standard graphs, the forcing circular number is identified. It is demonstrated that there exists a connected graph G such that f_g (G)=a and f_cr (G)=b for every integer a≥0, and b≥0.