Scaling limit of graph classes through split decomposition

TL;DR

This paper establishes that certain classes of random graphs, including distance-hereditary and 3-leaf power graphs, converge to Aldous' Brownian CRT under scaling, using split decomposition and analytic combinatorics.

Contribution

It proves the scaling limit of these graph classes is the Brownian CRT, extending understanding of their asymptotic geometric structure.

Findings

Distance-hereditary graphs converge to Brownian CRT

2-connected distance-hereditary graphs also converge to Brownian CRT

3-leaf power graphs have the same scaling limit

Abstract

We prove that Aldous' Brownian CRT is the scaling limit, with respect to the Gromov--Prokhorov topology, of uniform random graphs in each of the three following families of graphs: distance-hereditary graphs, $2$-connected distance-hereditary graphs and $3$-leaf power graphs. Our approach is based on the split decomposition and on analytic combinatorics.