On the Parameterized Complexity of Layered Graph Drawing  Vida Dujmovi´c  Michael R. Fellows  Matthew Kitching  Giuseppe Liotta † Catherine McCartin ‡ Naomi Nishimura § Prabhakar Ragde § Frances Rosamond  Sue Whitesides  David R. Wood ¶ 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 ad- mitting 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 so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers. 1 Introduction Layered graph drawing [War77,Car80,STT81] is a popular paradigm for drawing graphs, and has applications in visualization [DETT99], DNA mapping [WG86], and VLSI layout [Len90]. In a layered drawing of a graph, vertices are arranged in horizontal layers, and edges are routed as polygonal lines between distinct layers. For acyclic digraphs, it may be required that edges point downward. The quality of layered drawings is assessed in terms of criteria to be minimized, such as the number of edge crossings, the number of edges whose removal eliminates all crossings, the number of layers, the maximum number of layers an edge may cross, the total number of layers crossed by edges, and the maximum number of vertices in one layer. Unfortunately, the question of whether a graph G can be drawn in two layers with at most k crossings, where k is part of the input, is NP-complete [EW94b,GJ83], as is the question of whether r or fewer edges can be removed from G so that the remaining graph has a crossing-free drawing on two layers [TKY77,EW94a]. Both problems remain NP- complete when the permutation of vertices in one of the layers is given [EW94b,EW94a].  Research initiated at the International Workshop on Fixed Parameter Tractability in Graph Drawing, Bel- lairs Research Institute of McGill University, Holetown, Barbados, Feb. 9-16, 2001, organized by S. White- sides. Contact author: P. Ragde, Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 2P9, e-mail plragde@uwaterloo.ca. Research of Canada-based authors is supported by NSERC; research of Quebec-based authors is also supported by a grant from FCAR. Research of D. R. Wood completed while visiting McGill University. Research of G. Liotta supported by CNR and MURST.  Department of Mathematics and Statistics, McGill University, Canada.  School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, Australia. † Dipartimento di Ingegneria Elettronica e dell’Informazione, Universit`a degli Studi di Perugia, Italy. ‡ Institute of Information Science and Technology, Massey University, New Zealand. § School of Computer Science, University of Waterloo, Canada. ¶ Departament de Matem`atica Aplicada II, Universitat Polit`ecnica de Catalunya, Barcelona, Spain