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 a large number of 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 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.

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 ·6 k +|G|) algorithm for 2-Layer Planarization and an O(3 k ·|G|) algorithm for 1-Layer Planarization, thus achieving linear time for fixed k.

A bipartite graph is biplanar if the vertices can be placed on two parallel lines in the plane such that there are no edge crossings when edges are drawn as straight-line segments connecting vertices on one line to vertices on the other line. We study two problems: • 2-Layer Planarization: can k edges be deleted from a given graph G so that the remaining graph is biplanar? • 1-Layer Planarization: same question, but the order of the vertices on one layer is fixed. Improving on earlier works of Dujmović et al. (Proc. Graph Drawing GD 2001, pp. 1–15, 2002), we solve the 2-Layer Planarization problem in O(k 2 · 5.1926 k + |G|) time and the 1-Layer Planarization problem in O(k 3 · 2.5616 k + |G | 2) time. Moreover, we derive a small problem kernel for 1-Layer Planarization.

In this paper, we study a crossing minimization problem in a radial drawing of a graph. Radial drawings have strong application in social network visualization, for example, displaying centrality indices of actors [20]. The main problem of this paper is called the one-sided radial crossing minimization between two concentric circles, named orbits, where the positions of vertices in the outer orbit are fixed. The main task of the problem is to find the vertex ordering of the free orbit and edge routing between two orbits in order to minimize the number of edge crossings. The problem is known to be NP-hard [1], and the first polynomial time 15-approximation algorithm was presented in [9]. In this paper, we present a new 3α-approximation algorithm for the case when the free orbit has no leaf vertex, where α represents the approximation ratio of the one-sided crossing minimization problem in a horizontal drawing. Using the best known result of α =1.4664 [13], our new algorithm can achieve 4.3992-approximation. Submitted:

In this paper, we study a crossing minimization problem in a radial drawing of a graph. Radial drawings have strong application in social network visualization, for example, displaying centrality indices of actors [20]. The main problem of this paper is called the one-sided radial crossing minimization between two concentric circles, named orbits, where the positions of vertices in the outer orbit are fixed. The main task of the problem is to find the vertex ordering of the free orbit and edge routing between two orbits in order to minimize the number of edge crossings. The problem is known to be NP-hard [1], and the first polynomial time 15-approximation algorithm was presented in [9]. In this paper, we present a new 3α-approximation algorithm for the case when the free orbit has no leaf vertex, where α represents the approximation ratio of the one-sided crossing minimization problem in a horizontal drawing. Using the best known result of α = 1.4664 [13], our new algorithm can achieve 4.3992-approximation.