*NC* algorithms for partitioning planar graphs into induced forests and approximating NP-hard problems

## Abstract

It is well known that the vertex set of every planar graph can be partitioned into three subsets each of which induces a forest. Previously, there has been no *NC* algorithm for computing such a partition. In this paper, we design an optimal *NC* algorithm for computing such a partition for a given planar graph. It runs in *O*(log *n* log^{*}*n*) time using *O*(*n*/(log *n* log^{*}*n*)) processors on an EREW PRAM. This algorithm implies *optimal NC* approximation algorithms for many *NP*-hard maximum induced subgraph problems on planar graphs with a performance ratio of 3. We also present optimal *NC* algorithms for partitioning the vertex set of a given *K*_{4}-free or *K*_{2, 3}-free graph into two subsets each of which induces a forest. As consequences,we obtain *optimal NC* algorithms for 4-coloring *K*_{4}-free or *K*_{2, 3}-free graphs which are previously unknown to our knowledge.

## Keywords

Planar Graph Parallel Algorithm Performance Ratio Adjacency List Hamiltonian Circuit## Preview

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