Based on graph separator theorems, we develop a new simulation technique which allows us to resolve several open problems for on-line computations: (1)One tape can nondeterministically simulate two nondeterministic pushdown stores in 0( n 1.5√log n) time. Together with the Ω(n 1.5/tr log n) lower bound, this solves the open problem 1 in P. Duris et al. (“Proceedings, 15th ACM Symp. on Theory of Computing, 1983,” pp. 127–132) for the one-tape versus two pushdown store case. It also disproves the conjectured Ω( n 2) lower bound which holds in the deterministic case (W. Maass, Trans. Amer. Math. Soc. 292 (1985), 675–693; M. Li and P. B. M. Vitanyi, Inform. and Computation, in press.) (2) The languages actually used by Maass and Freivalds ( Inform. Process. 77 (1977), 839–842) can be accepted in O( n 2log log n/√log n) and O( n 1.5/ trlog n) time by a one-tape TM, respectively. Therefore they cannot be used to obtain the Ω(n 2) lower bound for one tape nondeterministically simulating two tapes. The algorithm depends on a new graph separator theorem. (3) Three pushdown stores are better than two pushdown stores for nondeterministic machines. This anwwers a rather old open problem of R. V. Book and S. A. Greibach ( Math. Systems Theory 4 (1970), 97–111) and P. Duris and Z. Galil ( in “Proceedings, 14th ACM Symp. on Theory of Comput., 1982,” pp. 1–7). (4) One tape can nondeterministically simulate one nondeterministic queue in 0( n 1.5√log n) time. This disproves the conjectured Ω(n 2) lower bound which holds in the deterministic case.