Parallel Planar Subgraph Isomorphism and Vertex Connectivity

TL;DR

This paper introduces the first parallel fixed-parameter algorithm for subgraph isomorphism in planar and minor-closed graphs, achieving near-linear work and poly-logarithmic depth, and also addresses vertex connectivity testing efficiently.

Contribution

It presents a novel parallel algorithm for subgraph isomorphism in planar and minor-closed graphs with low depth and near-linear work, and applies it to vertex connectivity testing.

Findings

Achieves near-linear work for subgraph isomorphism in planar graphs.

Provides a parallel algorithm for vertex connectivity with poly-logarithmic depth.

First known sub-quadratic work bound for connectivity in planar graphs.

Abstract

We present the first parallel fixed-parameter algorithm for subgraph isomorphism in planar graphs, bounded-genus graphs, and, more generally, all minor-closed graphs of locally bounded treewidth. Our randomized low depth algorithm has a near-linear work dependency on the size of the target graph. Existing low depth algorithms do not guarantee that the work remains asymptotically the same for any constant-sized pattern. By using a connection to certain separating cycles, our subgraph isomorphism algorithm can decide the vertex connectivity of a planar graph (with high probability) in asymptotically near-linear work and poly-logarithmic depth. Previously, no sub-quadratic work and poly-logarithmic depth bound was known in planar graphs (in particular for distinguishing between four-connected and five-connected planar graphs).