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 characterization 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{sup}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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