    # Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph 𝒫(n,q) which results from taking the union of two paths of length n≥ 1, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter 0<q(n)≤ 1. This random graph model, the tangled path, goes through an evolution: if q is close to 0 the graph bears resemblance to a path and as q tends to 1 it becomes an expander. In an effort to understand the evolution of 𝒫(n,q) we determine the treewidth and cutwidth of 𝒫(n,q) up to log factors for all q. We also show that the property of having a separator of size one has a sharp threshold. In addition, we prove bounds on the diameter, and vertex isoperimetric number for specific values of q.

## Authors

02/15/2022

### The treewidth and pathwidth of graph unions

For two graphs G_1 and G_2 on the same vertex set [n]:={1,2, …, n}, and ...
12/19/2017

### Transversals of Longest Paths

Let (G) be the minimum cardinality of a set of vertices that intersects ...
01/06/2020

### (Theta, triangle)-free and (even hole, K_4)-free graphs. Part 2 : bounds on treewidth

A theta is a graph made of three internally vertex-disjoint chordless p...
11/19/2017

### Approximating geodesics via random points

Given a `cost' functional F on paths γ in a domain D⊂R^d, in the form F(...
08/03/2020

### Shifting paths to avoidable ones

An extension of an induced path P in a graph G is an induced path P' suc...
05/10/2018

### Haplotype-aware graph indexes

The variation graph toolkit (VG) represents genetic variation as a graph...
06/21/2021

### Incentive-Compatible Kidney Exchange in a Slightly Semi-Random Model

Motivated by kidney exchange, we study the following mechanism-design pr...
##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.