Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs

Abstract

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}