On weighted sublinear separators

TL;DR

This paper investigates weighted graph separators, showing that the small set of exceptional vertices needed for balanced separators can be bounded by iterated logarithms, refining previous bounds.

Contribution

It improves the bound on the size of the exceptional vertex set from O(log |V(G)|) to a smaller iterated logarithm, enhancing understanding of weighted separator structures.

Findings

Bound on exceptional vertices set size is reduced to iterated logarithm scale.

Weighted balanced separators can be achieved with fewer exceptional vertices.

The result applies to graphs with sublinear separators in all subgraphs.

Abstract

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