Paper Info
Title
A Log-space Algorithm for Canonization of Planar Graphs
Abstract
Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.
Year
Venue
Keywords
2008
Clinical Orthopaedics and Related Research
computational complexity,graph isomorphism,upper bound,planar graph,space complexity
Field
DocType
Volume
Graph canonization,Block graph,Discrete mathematics,Combinatorics,Outerplanar graph,Biconnected component,Biconnected graph,Planar straight-line graph,Algorithm,SPQR tree,Planar graph,Mathematics
Journal
abs/0809.2
Citations 
PageRank 
References 
3
0.43
18
Authors
5
Name
Order
Citations
PageRank
Samir Datta120019.82
Nutan Limaye213420.59
Prajakta Nimbhorkar317015.04
Thomas Thierauf428833.59
Fabian Wagner582.59