# Parallel ear decomposition search (EDS) and st-numbering in graphs

## Abstract

The linear time serial algorithm for planarity testing of [LEC-67] uses the linear time serial algorithm of [ET-76] for *st*-numbering. This *st*-numbering algorithm is based on depth-first search (DFS). A known conjecture states that DFS, which is a key technique in designing serial algorithms, is not amenable to poly-log time parallelism using "around linearly" (or even polynomially) many processors. The first contribution of this paper is a general method for searching efficiently in parallel undirected graphs, called *ear-decomposition search* (EDS).

The second contribution demonstrates the applicability of this search method. We present an efficient parallel algorithm for *st*-numbering in a biconnected graph. The algorithm is quite subtle and runs in logarithmic time using a linear number of processors on a concurrent-read concurrent-write (CRCW) PRAM. An efficient parallel algorithm for the problem did not exist before. It was not even known to be in NC.

## Keywords

Parallel Algorithm Internal Vertex Numbering Algorithm Adjacency List Serial Algorithm## Preview

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