Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces

TL;DR

This paper extends linear-time algorithms for subgraph isomorphism from planar graphs to graphs on surfaces of bounded genus, introducing surface split decompositions to simplify and generalize previous methods.

Contribution

It introduces surface split decompositions for bounded genus graphs and generalizes existing algorithms for subgraph isomorphism, including counting and generating all subgraphs.

Findings

Linear-time algorithm for subgraph isomorphism on bounded genus graphs

Simplified algorithm and analysis for these graphs

Extended algorithm to count and generate all subgraphs, including disconnected P

Abstract

The Subgraph Isomorphism problem asks, given a host graph G on n vertices and a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P. The restriction of this problem to planar graphs has often been considered. After a sequence of improvements, the current best algorithm for planar graphs is a linear time algorithm by Dorn (STACS '10), with complexity $2^{O(k)} O(n)$. We generalize this result, by giving an algorithm of the same complexity for graphs that can be embedded in surfaces of bounded genus. At the same time, we simplify the algorithm and analysis. The key to these improvements is the introduction of surface split decompositions for bounded genus graphs, which generalize sphere cut decompositions for planar graphs. We extend the algorithm for the problem of counting and generating all subgraphs isomorphic to P, even for the case where P is disconnected. This answers an open question by Eppstein (SODA '95 / JGAA '99).