A Fixed-Parameter Approach to Two-Layer Planarization
Dujmović, Vida and Fellows, M. and Hallett, M. and Kitching, Matthew and Liotta, Giuseppe and McCartin, C. and Nishimura, N. and Ragde, P. and Rosamond, F. and Suderman, Matthew and Whitesides, Sue and Wood, David R. (2002) A Fixed-Parameter Approach to Two-Layer Planarization. In: Graph Drawing 9th International Symposium, GD 2001, September 23-26, 2001, Vienna, Austria , pp. 1-15 (Official URL: http://dx.doi.org/10.1007/3-540-45848-4_1).
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A bipartite graph is biplanar if the vertices can be placed on two parallel lines (layers) in the plane such that there are no edge crossings when edges are drawn straight. The 2-LAYER PLANARIZATION problem asks if k edges can be deleted from a given graph G so that the remaining graph is biplanar. This problem is NP-complete, as is the 1-LAYER PLANARIZATION problem in which the permutation of the vertices in one layer is fixed. We give the following fixed parameter tractability results: an O(k \centerdot 6^k+|G|) algorithm for 2-LAYER PLANARIZATION and an O(3^k\centerdot |G|) algorithm for 1-LAYER PLANARIZATION, thus achieving linear time for fixed k.
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