# Parallel complexity of lexicographically first problems for tree-structured graphs

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## Abstract

We study several P-complete graph problems and show that they are in NC if the input graphs are restricted to be tree-structured. These graphs are also known as partial k-trees, decomposable graphs or graphs of bounded tree width, and include outerplanar graphs, series-parallel graphs and Halin graphs. The particular problems investigated herein include the lexicographically first (l.f.) depth-first search tree and the l.f. maximal independent set. It is also shown that if a tree of faces of an outerplanar graph is given, then its dfs tree can be found in O(log_{2}n) time using O(n/log_{2}n) processors.

## Keywords

Decomposition Tree Outerplanar Graph Halin Graph Optimal Parallel Algorithm Decomposable Graph
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