Edge separators for quasi-binary trees

TL;DR

This paper studies how to partition quasi-binary trees into roughly equal weighted components by removing edges, providing bounds and optimality examples for such separators.

Contribution

It introduces methods for constructing balanced $k$-separators in quasi-binary trees with bounds on component weights, under certain total weight conditions.

Findings

Existence of $k$-separators with bounded component weights

Bounds are proven to be optimal in some cases

Applicable to vertex-weighted quasi-binary trees

Abstract

One wishes to remove $k-1$ edges of a vertex-weighted tree $T$ such that the weights of the $k$ induced connected components are approximately the same. How well can one do it ? In this paper, we investigate such $k$-separator for {\em quasi-binary} trees. We show that, under certain conditions on the total weight of the tree, a particular $k$-separator can be constructed such that the smallest (respectively the largest) weighted component is lower (respectively upper) bounded. Examples showing optimality for the lower bound are also given.