Weighted graph separators and their applications

Abstract

We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total sum of vertex costs at most c√∑v{small element of}V(G)(cost(V))2 and that this bound is optimal to within a constant factor. Moreover such a separator can be found in linear time. This theorem implies a variety of other separation results. We describe applications of our separator theorems to graph embedding problems, to graph pebbling, and to multi-commodity flow problems.