Abstract
Graph canonization is the problem of computing a unique representative, a canon, from the isomorphism class of a given graph. This implies that two graphs are isomorphic exactly if their canons are equal. We show that graphs of bounded tree width can be canonized by logarithmic-space (logspace) algorithms. This implies that the isomorphism problem for graphs of bounded tree width can be decided in logspace. In the light of isomorphism for trees being hard for the complexity class logspace, this makes the ubiquitous class of graphs of bounded tree width one of the few classes of graphs for which the complexity of the isomorphism problem has been exactly determined.
Highlights
The graph isomorphism problem – deciding whether two given graphs are the same up to renaming vertices – is one of the few fundamental problems in NP for which we neither know that it is solvable in polynomial-time nor that it is NP-complete
Since isomorphism is hard for NL [26], a deeper complexitytheoretic insight behind the polynomial-time algorithms for embeddable graphs is given by the fact that isomorphism for graphs embeddable into the plane [7] or a fixed surface [10] is in L
For every k ∈ N, there is a logspace-computable and isomorphism-invariant mapping that turns a graph G with tree width at most k into a tree decomposition D for G in which (1) subgraphs induced by the bags do not contain clique separators, and (2) adhesion sets are cliques
Summary
The graph isomorphism problem (isomorphism) – deciding whether two given graphs are the same up to renaming vertices – is one of the few fundamental problems in NP for which we neither know that it is solvable in polynomial-time nor that it is NP-complete. Since isomorphism is hard for NL (nondeterministic logarithmic space) [26], a deeper complexitytheoretic insight behind the polynomial-time algorithms for embeddable graphs is given by the fact that isomorphism for graphs embeddable into the plane [7] or a fixed surface [10] is in L (deterministic logarithmic space, called logspace). It has been an open question whether for graphs of bounded tree width the isomorphism problem can be solved in logspace. While providing us with logspace algorithms for ever larger classes of graphs, the general question remained open
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