On the Parameterized Complexity of Layered Graph Drawing

Authors:
Vida Dujmović;Michael R. Fellows;Matthew Kitching;Giuseppe Liotta;Catherine McCartin;Naomi Nishimura;Prabhakar Ragde;Frances Rosamond;Sue Whitesides;David R. Wood
Affiliations:
McGill University, Department of Mathematics and Statistics, H3A 2T5, Montreal, Quebec, Canada;The University of Newcastle, School of Electrical Engineering and Computer Science, H3A 2T5, Newcastle, Quebec, Australia;McGill University, Department of Mathematics and Statistics, H3A 2T5, Montreal, Quebec, Canada;Università degli Studi di Perugia, Dipartimento di Ingegneria Elettronica e dell’Informazione, H3A 2T5, Perugia, Quebec, Italy;Massey University, Institute of Information Science and Technology, H3A 2T5, Palmerston North, Quebec, New Zealand;University of Waterloo, School of Computer Science, N2L 2P9, Waterloo, Ontario, Canada;University of Waterloo, School of Computer Science, N2L 2P9, Waterloo, Ontario, Canada;The University of Newcastle, School of Electrical Engineering and Computer Science, H3A 2T5, Newcastle, Quebec, Australia;McGill University, Department of Mathematics and Statistics, H3A 2T5, Montreal, Quebec, Canada;Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada II, N2L 2P9, Barcelona, Ontario, Spain
Venue:
Algorithmica - Parameterized and Exact Algorithms
Year:
2008

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Theoretical Computer Science

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Abstract

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.