Thin Tree Position

TL;DR

This paper introduces a novel method for constructing and refining tree structures from weighted graphs, called thin tree positions, which facilitate effective graph partitioning with desirable properties, inspired by topological concepts.

Contribution

It presents a new algorithm for creating thin tree positions from graphs, extending topological ideas to improve graph partitioning techniques.

Findings

Thin tree positions enable better graph partitions.

The method is based on topological notions of thin position.

The algorithm produces partitions with advantageous properties.

Abstract

We introduce a method for creating a special type of tree, called a tree position, from a weighted graph. Leaves of the tree correspond to vertices of the original graph, and the tree edges contain information which can be used to partition these vertices. By repeatedly applying reducing operations to the tree position we arrive at a special type of tree position we call thin, and we show that partitions arising from thin tree positions have especially nice properties. The algorithm is based the topological notion of thin position for knots and 3-manifolds and builds on the previously defined idea of a Topological Intrinsic Lexicographic Order (TILO).