DeepAI

# Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph π«(n,q) which results from taking the union of two paths of length nβ₯ 1, where the vertices of one of the paths have been relabelled according to a Mallows permutation with real parameter 0<q(n)β€ 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 π«(n,q) we determine the treewidth and cutwidth of π«(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.

• 11 publications
• 18 publications
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• 13 publications
β 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, β¦, n}, and ...
β 12/19/2017

### Transversals of Longest Paths

Let (G) be the minimum cardinality of a set of vertices that intersects ...
β 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...
β 11/19/2017

### Approximating geodesics via random points

Given a `cost' functional F on paths Ξ³ in a domain DβR^d, in the form F(...
β 08/22/2020

### Structural Parameterizations of Tracking Paths Problem

Given a graph G with source and destination vertices s,tβ V(G) respectiv...
β 05/10/2018

### Haplotype-aware graph indexes

The variation graph toolkit (VG) represents genetic variation as a graph...
β 12/12/2019

### Short simplex paths in lattice polytopes

We consider the problem of optimizing a linear function over a lattice p...