Graphs that Admit Right Angle Crossing Drawings

Arikushi, Karin; Fulek, Radoslav; Keszegh, Balázs; Morić, Filip; Tóth, Csaba D.

doi:10.1007/978-3-642-16926-7_14

Cited by 13 publications

(22 citation statements)

References 9 publications

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“…Recently 1-planar graphs and RAC graphs have been extensively studied [2,8,10,16]. Pach and Toth proved that 1-planar graphs with n vertices have at most 4n − 8 edges [14], whereas Didimo et al showed that a RAC graph with n > 3 vertices has at most 4n − 10 edges [7].…”

Section: Introductionmentioning

confidence: 99%

Bar 1-Visibility Drawings of 1-Planar Graphs

et al. 2014

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Abstract. A bar 1-visibility drawing of a graph G is a drawing of G 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 G is bar 1-visible if G has a bar 1-visibility drawing. A graph G is 1-planar if G 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.

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Section: Introductionmentioning

confidence: 99%

Bar 1-Visibility Drawings of 1-Planar Graphs

et al. 2014

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“…A recent paper of Arikushi et al [8] considers a similar question. They consider a drawing of a graph G on n vertices in the plane where each edge is represented by a polygonal arc joining its two respective vertices.…”

Section: Previous Workmentioning

confidence: 99%

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Dujmović¹,

Gudmundsson²,

Morin³

et al. 2011

Chicago J. of Theoretical Comp. Sci.

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A geometric graph G is an α angle crossing (αAC) graph if every pair of crossing edges in G cross at an angle of at least α. The concept of right angle crossing (RAC) graphs (α = π/2) was recently introduced by Didimo et al. [10]. It was shown that any RAC graph with n vertices has at most 4n − 10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n − 10 edges. In this paper, we give upper and lower bounds for the number of edges in αAC graphs for all 0 < α < π/2.

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“…Assume 3 , there are at least two edges incident to v 3 that cross the boundary of f 2 . If both these edges cross edge (u, z), then the same argument as in the previous case applies.…”

Section: Lemma 6 Every Rac Drawing Of a Maximally Dense Rac Graph Ismentioning

confidence: 99%

“…If both these edges cross edge (u, z), then the same argument as in the previous case applies. If one of these edges crosses (v, z), it must also cross (w, z) to reach any one of 3 , there is an edge of this cycle crossing (u, z), one crossing (v, z), and one crossing (w, z). Again by Lemma 6, neither (u, z), nor (v, z), nor (w, z) can be crossed by any other edge.…”

Section: Lemma 6 Every Rac Drawing Of a Maximally Dense Rac Graph Ismentioning

confidence: 99%

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Right Angle Crossing Graphs and 1-Planarity

Eades

Liotta

2012

Graph Drawing

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A Right Angle Crossing Graph (also called RAC graph for short) is a graph that has a straight-line drawing where any two crossing edges are orthogonal to each other. A 1-planar graph is a graph that has a drawing where every edge is crossed at most once. We study the relationship between RAC graphs and 1-planar graphs in the extremal case that the RAC graphs have as many edges as possible. It is known that a maximally dense RAC graph with n > 3 vertices has 4n − 10 edges. We show that every maximally dense RAC graph is 1-planar. Also, we show that for every integer i such that i ≥ 0, there exists a 1-planar graph with n = 8 + 4i vertices and 4n − 10 edges that is not a RAC graph.

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Graphs that Admit Right Angle Crossing Drawings

Cited by 13 publications

References 9 publications

Bar 1-Visibility Drawings of 1-Planar Graphs

Bar 1-Visibility Drawings of 1-Planar Graphs

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Right Angle Crossing Graphs and 1-Planarity

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