ON NONTRIVIAL SEPARATORS FOR K-PAGE GRAPHS AND SIMULATIONS BY NONDETERMINISTIC ONE-TAPE TURING-MACHINES

We show that the following statements are equivalent: Statement 1: 3-pushdown graphs have sublinear separators; Statement 1*: k-page graphs have sublinear separators; and Statement 2: A one-tape nondeterministic Turing machine can simulate a two-tape machine in subquadratic time. None of the statements is known to be true or false at present. However, our proof of equivalence is quantitative. It relates exactly the separator size of the two kinds of graphs to the running time of the simulation in Statement 2. Using this equivalence we derive several graph-theoretic corollaries. Our results may constitute the first example where a graph problem is shown to be equivalent to a problem in computational complexity. Using the equivalence we prove an almost linear lower bound for the size of separators for 3-pushdown graphs and an almost quadratic lower bound for simulating two-tape nondeterministic Turing machines by one-tape machines.