# Space Efficient Algorithms for Series-Parallel Graphs

## Abstract

The subclass of directed *series-parallel graphs* plays an important role in computer science. To determine whether a graph is series-parallel is a well studied problem in algorithmic graph theory. Fast sequential and parallel algorithms for this problem have been developed in a sequence of papers. For series-parallel graphs methods are also known to solve the reachability and the decomposition problem time efficiently. However, no dedicated results have been obtained for the space complexity of these problems - the topic of this paper. For this special class of graphs, we develop deterministic algorithms for the *recognition, reachability, decomposition* and the *path counting problem* that use only logarithmic space. Since for arbitrary directed graphs reachability and path counting are believed not to be solvable in log-space the main contribution of this work are novel deterministic path finding routines that work correctly in series-parallel graphs, and a characterisation of series-parallel graphs by forbidden subgraphs that can be tested space-efficiently. The space bounds are best possible, i.e. the decision problems is shown to be *L*-complete with respect to *AC* ^{0}-reductions, and they have also implications for the parallel time complexity of series-parallel graphs. Finally, we sketch how these results can be generalised to extension of the series-parallel graph family: to graphs with multiple sources or multiple sinks and to the class of *minimal vertex series-parallel graphs*.

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