Local 2-separators

TL;DR

This paper introduces 'Local Separators' in graphs to extend sparse graph theory to large networks, providing new decomposition theorems and characterizations relevant for analyzing complex graph structures.

Contribution

It presents the concept of local 2-separators, a decomposition theorem similar to the 2-separator theorem, and characterizes graphs without bounded wheel subdivisions.

Findings

Decomposition theorem for graphs using local 2-separators

Characterization of graphs with no bounded wheel subdivision

Extension of sparse graph theory to large networks

Abstract

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2-separators analogous to the 2-separator theorem; 2. an exact characterisation of graphs with no bounded subdivision of a wheel.