Tangled Paths: A Random Graph Model from Mallows Permutations

08/10/2021
βˆ™
by   Jessica Enright, et al.
βˆ™
0
βˆ™

We introduce the random graph 𝒫(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)≀ 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
βˆ™ 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 ...
research
βˆ™ 12/19/2017

Transversals of Longest Paths

Let (G) be the minimum cardinality of a set of vertices that intersects ...
research
βˆ™ 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...
research
βˆ™ 11/19/2017

Approximating geodesics via random points

Given a `cost' functional F on paths Ξ³ in a domain DβŠ‚R^d, in the form F(...
research
βˆ™ 07/04/2023

On the structural and combinatorial properties in 2-swap word permutation graphs

In this paper, we study the graph induced by the 2-swap permutation on w...
research
βˆ™ 05/10/2018

Haplotype-aware graph indexes

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

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

Motivated by kidney exchange, we study the following mechanism-design pr...

Please sign up or login with your details

Forgot password? Click here to reset