9,018 research outputs found

    On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

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    Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm to test whether a given graph is maximal outer-fan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-hard, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given

    On the Number of Edges of Fan-Crossing Free Graphs

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    A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges g,e1,...ekg,e_1,...e_k, such that e1,e2,...eke_1,e_2,...e_k have a common endpoint and gg crosses all eie_i. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties

    Max-Cut and Max-Bisection are NP-hard on unit disk graphs

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    We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk graphs. We also show that λ\lambda-precision graphs are planar for λ\lambda > 1 / \sqrt{2}$

    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

    Recognizing and Drawing IC-planar Graphs

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    IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph GG with nn vertices, we present an O(n)O(n)-time algorithm that computes a straight-line drawing of GG in quadratic area, and an O(n3)O(n^3)-time algorithm that computes a straight-line drawing of GG with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

    Bar 1-Visibility Drawings of 1-Planar Graphs

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    A bar 1-visibility drawing of a graph GG is a drawing of GG where each vertex is drawn as a horizontal line segment called a bar, each edge is drawn as a vertical line segment where the vertical line segment representing an edge must connect the horizontal line segments representing the end vertices and a vertical line segment corresponding to an edge intersects at most one bar which is not an end point of the edge. A graph GG is bar 1-visible if GG has a bar 1-visibility drawing. A graph GG is 1-planar if GG has a drawing in a 2-dimensional plane such that an edge crosses at most one other edge. In this paper we give linear-time algorithms to find bar 1-visibility drawings of diagonal grid graphs and maximal outer 1-planar graphs. We also show that recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs are bar 1-visible graphs.Comment: 15 pages, 9 figure

    Straight-line Drawability of a Planar Graph Plus an Edge

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    We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The characterization enables a linear-time testing algorithm to determine whether an almost-planar graph admits a straight-line drawing, and a linear-time drawing algorithm that constructs such a drawing, if it exists. We also show that some almost-planar graphs require exponential area for a straight-line drawing
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