We consider straight line drawings of a planar graph G with possible edge crossings. The untangling problem is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let fix (G) denote the maximum number of vertices that can be left fixed in the worst case. In the allocation problem, we are given a planar graph G on n vertices together with an n-point set X in the plane and have to draw G without edge crossings so that as many vertices as possible are located in X. Let fit(G) denote the maximum number of points fitting this purpose in the worst case. As fix (G) ≤ fit(G), we are interested in upper bounds for the latter and lower bounds for the former parameter.For each ǫ > 0, we construct an infinite sequence of graphs with fit(G) = O(n σ+ǫ ), where σ < 0.99 is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. To the best of our knowledge, this is the first example of graphs with fit(G) = o(n). On the other hand, we prove that fix (G) ≥ n/30 for all G with tree-width at most 2. This extends the lower bound obtained by Goaoc et al. [Discrete and Computational Geometry 42:542-569 (2009)] for outerplanar graphs.Our upper bound for fit (G) is based on the fact that the constructed graphs can have only few collinear vertices in any crossing-free drawing. To prove the lower bound for fix (G), we show that graphs of tree-width 2 admit drawings that have large sets of collinear vertices with some additional special properties.1
Abstract. We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows.-We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. -We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). -We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.
It is well known that any graph admits a crossing-free straight-line drawing in R 3 and that any planar graph admits the same even in R 2 . For a graph G and d ∈ {2, 3}, let ρ 1 d (G) denote the minimum number of lines in R d that together can cover all edges of a drawing of G. For d = 2, G must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. *
Abstract. Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an xmonotone curve that goes from left to right. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. Our main results are as follows. We show that minimizing the number of block crossings is NP-hard, and we develop, for meetings of bounded size, a constant-factor approximation. We also present two fixed-parameter algorithms and, for meetings of size 2, a greedy heuristic that we evaluate experimentally.
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