Efficient parallel algorithms for series parallel graphs

We present efficient parallel algorithms for solving three problems for series parallel graphs: 3-coloring, depth-first spanning tree, and breadth-first spanning tree. If the input is given by the decomposition tree, the first two problems can be solved in O(log n) time with O( n log n) processors, the last problem can be solved in O(log n log log n) time with O(n) processors. We also present a parallel algorithm for recognizing series parallel graphs and constructing decomposition trees. This algorithm takes O(log² n + log m) time with O(n + m) processors, where n (m) is the number of vertices (edges) in the graph.