Edge bundling reduces the visual clutter in a drawing of a graph by uniting the edges into bundles. We propose a method of edge bundling drawing each edge of a bundle separately as in metro-maps and call our method ordered bundles. To produce aesthetically looking edge routes it minimizes a cost function on the edges. The cost function depends on the ink, required to draw the edges, the edge lengths, widths and separations. The cost also penalizes for too many edges passing through narrow channels by using the constrained Delaunay triangulation. The method avoids unnecessary edge-node and edge-edge crossings. To draw edges with the minimal number of crossings and separately within the same bundle we develop an efficient algorithm solving a variant of the metro-line crossing minimization problem. In general, the method creates clear and smooth edge routes giving an overview of the global graph structure, while still drawing each edge separately and thus enabling local analysis.
The metro-line crossing minimization (MLCM) problem was recently introduced as a response to the problem of drawing metro maps or public transportation networks, in general. According to this problem, we are given a planar, embedded graph G = (V ,E ) and a set L of simple paths on G , called lines . The main task is to place the lines on G , so that the number of crossings among pairs of lines is minimized. Our main contribution is two polynomial time algorithms. The first solves the general case of the MLCM problem, where the lines that traverse a particular vertex of G are allowed to use any side of it to either "enter" or "exit", assuming that the endpoints of the lines are located at vertices of degree one. The second one solves more efficiently the restricted case, where only the left and the right side of each vertex can be used. To the best of our knowledge, this is the first time where the general case of the MLCM problem is solved. Previous work was devoted to the restricted case of the MLCM problem under the additional assumption that the endpoints of the lines are either the topmost or the bottommost in their corresponding vertices, i.e., they are either on top or below the lines that pass through the vertex. Even for this case, we improve a known result of from O (|E |5 2|L |3) to O (|V |(|E | + |L |)).  In this paper we consider a problem that occurs when drawing public transportation networks. Given an embedded graph G = (V, E) (e.g. the railroad network) and a set H of paths in G (e.g. the train lines), we want to draw the paths along the edges of G such that they cross each other as few times as possible. For aesthetic reasons we insist that the relative order of the paths that traverse a vertex does not change within the area occupied by the vertex. We prove that the problem, which is known to be NP-hard, can be rewritten as an integer linear program that finds the optimal solution for the problem. In the case when the order of the endpoints of the paths is fixed we prove that the problem can be solved in polynomial time. This improves a recent result by (2007).