# Pathwidth, trees, and random embeddings

## TL;DR

This paper proves that shortest-path metrics on graphs with bounded pathwidth can be embedded into random trees with bounded distortion, and uses this to confirm a conjecture about flow/cut gaps in certain graph families.

## Contribution

It establishes a new embedding result for graphs of bounded pathwidth and applies it to prove a conjecture on flow/cut gaps for minor-closed graph families.

## Key findings

- Shortest-path metrics on pathwidth-$k$ graphs embed into random trees with bounded distortion.
- Confirms the flow/cut gap conjecture for minor-closed families excluding all trees.
- Provides a new tool for analyzing metric embeddings in graph theory.

## Abstract

We prove that, for every $k=1,2,...,$ every shortest-path metric on a graph of pathwidth $k$ embeds into a distribution over random trees with distortion at most $c$ for some $c=c(k)$. A well-known conjecture of Gupta, Newman, Rabinovich, and Sinclair states that for every minor-closed family of graphs $F$, there is a constant $c(F)$ such that the multi-commodity max-flow/min-cut gap for every flow instance on a graph from $F$ is at most $c(F)$. The preceding embedding theorem is used to prove this conjecture whenever the family $F$ does not contain all trees.