Problem: Let S be a minimum total (connected) open monophonic set of G. A subset T of S is called a forcing subset for S if S is the unique minimum total (connected) open monophonic set containing T. A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing total (connected)open monophonic number of S denoted by f_tom(S) ( f_com(S) )is the cardinality of a minimum forcing subset of S. The forcing total (connected) open monophonic number of G denoted by f_tom(G) (f_com(G)) is f_tom(G) (f_com(G) ) = min{f_tom(S)} (min{f_com(S)}), where the minimum is taken over all minimum total open monophonic sets in G. Findings: We determine bounds for it and characterize graphs which realize these bounds. Forcing total (connected) open monophonic number of certain standard graphs are found. Also, we proved that following results: (i) for any positive integers a, n with 0 ≤ a≤ n− 4, there exists a connected graph G of order n such that f_tom(G) = 0 or 1, or 2 and om_t(G) = a. (ii) for any integer b with b ≥ 6, there exists a connected graph G such that f_com(G) =  0 or 1 and om_c(G) = b.

2000 Mathematics Subject Classification 05C12,05C70.