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.
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© 1995 Springer-Verlag Berlin Heidelberg
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Chen, ZZ., He, X. (1995). NC algorithms for partitioning planar graphs into induced forests and approximating NP-hard problems. In: Nagl, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 1995. Lecture Notes in Computer Science, vol 1017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60618-1_82
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DOI: https://doi.org/10.1007/3-540-60618-1_82
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