On classes of graphs with strongly sublinear separators

TL;DR

This paper studies graph classes with strongly sublinear separators, providing properties and an efficient algorithm to approximate membership in such classes, which helps in understanding their structure and recognition.

Contribution

It introduces an approximate polynomial-time algorithm to determine if a graph belongs to classes with strongly sublinear separators, advancing understanding of their properties.

Findings

Existence of an approximate algorithm for class membership

Polynomial-time determination of graph class membership

Characterization of graphs with strongly sublinear separators

Abstract

For real numbers c,epsilon>0, let G_{c,epsilon} denote the class of graphs G such that each subgraph H of G has a balanced separator of order at most c|V(H)|^{1-epsilon}. A class of graphs has strongly sublinear separators if it is a subclass of G_{c,epsilon} for some c,epsilon>0. We investigate properties of such graph classes, leading in particular to an approximate algorithm to determine membership in G_{c,epsilon}: there exist c'>0 such that for each input graph G, this algorithm in polynomial time determines either that G belongs to G_{c',epsilon^2/160}, or that G does not belong to G_{c,epsilon}.