A dynamic data structure for counting subgraphs in sparse graphs

TL;DR

This paper introduces a dynamic data structure that efficiently maintains and counts subgraphs within sparse graphs, supporting rapid updates and queries, especially effective for graphs with bounded expansion.

Contribution

It presents a novel dynamic data structure capable of counting subgraphs in sparse graphs with constant query time and polylogarithmic update time for graphs with bounded expansion.

Findings

Constant time subgraph counting queries

Polylogarithmic amortized update time for bounded expansion graphs

Applicable to classes like planar graphs and graphs with bounded average degree

Abstract

We present a dynamic data structure representing a graph G, which allows addition and removal of edges from G and can determine the number of appearances of a graph of a bounded size as an induced subgraph of G. The queries are answered in constant time. When the data structure is used to represent graphs from a class with bounded expansion (which includes planar graphs and more generally all proper classes closed on topological minors, as well as many other natural classes of graphs with bounded average degree), the amortized time complexity of updates is polylogarithmic.