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We give a new upper bound for the Group and Quasigroup Isomorphism problems when the input structures are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(\log\log n)$ depth and $O(\log^2 n)$ nondeterministic bits, where $n$ is the number of group elements. This improves the existing upper bound from \cite{Wolf 94} for the problems. In the previous upper bound the circuits have bounded fan-in but depth $O(\log^2 n)$ and also $O(\log^2 n)$ nondeterministic bits. We then prove that the kind of circuits from our upper bound cannot compute the Parity function. Since Parity is AC0 reducible to Graph Isomorphism, this implies that Graph Isomorphism is strictly harder than Group or Quasigroup Isomorphism under the ordering defined by AC0 reductions.

The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width are known to be solvable in polynomial time~\cite{Bo90},\cite{YBFT}.We give restricted space algorithms for these problems proving the following results: \begin{itemize} \item Isomorphism for bounded tree distance width graphs is in \Log\ and thus complete for the class. We also show that for this kind of graphs a canon can be computed within logspace. \item For bounded treewidth graphs, when both input graphs are given together with a tree decomposition, the problem of whether there is an isomorphism which respects the decompositions (i.e.\ considering only isomorphisms mapping bags in one decomposition blockwise onto bags in the other decomposition) is in \Log. \item For bounded treewidth graphs, when one of the input graphs is given with a tree decomposition the isomorphism problem is in \LogCFL. \item As a corollary the isomorphism problem for bounded treewidth graphs is in \LogCFL. This improves the known \TCone\ upper bound for the problem given by Grohe and Verbitsky~\cite{GV06}. \end{itemize}

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. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and important special case by presenting an upper bound that matches the known log-space hardness [JKMT03]. In fact, we show the formally stronger result that planar graph canonization is in log-space. This improves the previously known upper bound of AC1 [MR91]. 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 log-space reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in log-space by [DLN08]. 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.

Graph isomorphism is an important and widely studied computational problem with a yet unsettled complexity. However, the exact complexity is known for isomorphism of various classes of graphs. Recently, \cite{DLNTW09} proved that planar isomorphism is complete for log-space. We extend this result %of \cite{DLNTW09} further to the classes of graphs which exclude $K_{3,3}$ or $K_5$ as a minor, and give a log-space algorithm. Our algorithm decomposes $K_{3,3}$ minor-free graphs into biconnected and those further into triconnected components, which are known to be either planar or $K_5$ components \cite{Vaz89}. This gives a triconnected component tree similar to that for planar graphs. An extension of the log-space algorithm of \cite{DLNTW09} can then be used to decide the isomorphism problem. For $K_5$ minor-free graphs, we consider $3$-connected components. These are either planar or isomorphic to the four-rung mobius ladder on $8$ vertices or, with a further decomposition, one obtains planar $4$-connected components \cite{Khu88}. We give an algorithm to get a unique decomposition of $K_5$ minor-free graphs into bi-, tri- and $4$-connected components, and construct trees, accordingly. Since the algorithm of \cite{DLNTW09} does not deal with four-connected component trees, it needs to be modified in a quite non-trivial way.

The isomorphism problem for planar graphs is known to be efficiently solvable. For planar 3-connected graphs, the isomorphism problem can be solved by efficient parallel algorithms, it is in the class $AC^1$. In this paper we improve the upper bound for planar 3-connected graphs to unambiguous logspace, in fact to $UL cap coUL$. As a consequence of our method we get that the isomorphism problem for oriented graphs is in $NL$. We also show that the problems are hard for $L$.