1,251 research outputs found

    Aligned Drawings of Planar Graphs

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    Let GG be a graph that is topologically embedded in the plane and let A\mathcal{A} be an arrangement of pseudolines intersecting the drawing of GG. An aligned drawing of GG and A\mathcal{A} is a planar polyline drawing Γ\Gamma of GG with an arrangement AA of lines so that Γ\Gamma and AA are homeomorphic to GG and A\mathcal{A}. We show that if A\mathcal{A} is stretchable and every edge ee either entirely lies on a pseudoline or it has at most one intersection with A\mathcal{A}, then GG and A\mathcal{A} have a straight-line aligned drawing. In order to prove this result, we strengthen a result of Da Lozzo et al., and prove that a planar graph GG and a single pseudoline L\mathcal{L} have an aligned drawing with a prescribed convex drawing of the outer face. We also study the less restrictive version of the alignment problem with respect to one line, where only a set of vertices is given and we need to determine whether they can be collinear. We show that the problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    Planar Drawings of Fixed-Mobile Bigraphs

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    A fixed-mobile bigraph G is a bipartite graph such that the vertices of one partition set are given with fixed positions in the plane and the mobile vertices of the other part, together with the edges, must be added to the drawing. We assume that G is planar and study the problem of finding, for a given k >= 0, a planar poly-line drawing of G with at most k bends per edge. In the most general case, we show NP-hardness. For k=0 and under additional constraints on the positions of the fixed or mobile vertices, we either prove that the problem is polynomial-time solvable or prove that it belongs to NP. Finally, we present a polynomial-time testing algorithm for a certain type of "layered" 1-bend drawings

    On Hardness of the Joint Crossing Number

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    The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar

    Geometric Graph Drawing Algorithms - Theory, Engineering and Experiments

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    New Parameters for Beyond-Planar Graphs

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    Parameters for graphs appear frequently throughout the history of research in this field. They represent very important measures for the properties of graphs and graph drawings, and are often a main criterion for their classification and their aesthetic perception. In this direction, we provide new results for the following graph parameters: – The segment complexity of trees; – the membership of graphs of bounded vertex degree to certain graph classes; – the maximal complete and complete bipartite graphs contained in certain graph classes beyond-planarity; – the crossing number of graphs; – edge densities for outer-gap-planar graphs and for bipartite gap-planar graphs with certain properties; – edge densities and inclusion relationships for 2-layer graphs, as well as characterizations for complete bipartite graphs in the 2-layer setting
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