For a connected graph G = (V, E) of order at least two, subset T minimum total monophonic set S is forcing for if the unique containing . A cardinality S. The number ftm(S) in ftm(G) min{ftm(S)}, where taken over all sets G. We determine bounds it and find certain classes graphs. It shown that every pair a, b positive integers with 0 ≤ < ≥ a+4, there exists such mt(G) b.

The minimum forcing number of a graph G is the smallest edges simultaneously contained in unique perfect matching G. Zhang et al. (2010) claimed that any fullerene was bounded below by 3. However, we find there exists exactly one exceptional F24 with 2. In this paper, characterize all fullerenes 3 construction approach. This also solves an open problem proposed We except for F24, anti-forcing 4...