Abstract
Consider a graph (Formula presented.) with an assignment of costs to vertices. Even if (Formula presented.) and all its subgraphs admit balanced separators of sublinear size, (Formula presented.) may only admit a balanced separator of sublinear cost after deleting a small set (Formula presented.) of exceptional vertices.
We improve the bound on (Formula presented.) from (Formula presented.) to (Formula presented.), for any fixed number of iterations of the logarithm.