The Straight-Line RAC Drawing Problem Is NP-Hard

Argyriou, Evmorfia N.; Bekos, Michael A.; Symvonis, Antonios

doi:10.1007/978-3-642-18381-2_6

Cited by 26 publications

(33 citation statements)

References 16 publications

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“…For 1-planarity this was proved by Grigoriev and Bodlaender [24] and by Korzhik and Mohar [28], and improved to hold for graphs of bounded bandwidth, pathwidth, or treewidth [4], for near planar graphs [15], and for 3-connected 1-planar graphs with a given rotation system [3]. Moreover, the recognition of right angle crossing graphs (RAC) [1] and of fan-planar graphs [5,6] is N P-hard. On the other hand, Eades et al [21] introduced a linear time testing algorithm for (planar) maximal 1-planar graphs that are given with a rotation system.…”

Section: Introductionmentioning

confidence: 90%

Recognizing Optimal 1-Planar Graphs in Linear Time

Brandenburg

2016

Algorithmica

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A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n − 8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our algorithm implements a graph reduction system with two rules, which can be used to reduce every optimal 1-planar graph to an irreducible extended wheel graph. The graph reduction system is non-deterministic, constraint, and non-confluent.

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Section: Introductionmentioning

confidence: 90%

Recognizing Optimal 1-Planar Graphs in Linear Time

Brandenburg

2016

Algorithmica

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“…Proof of Theorem 1.3: Let G = (V, E) be a topological graph on n vertices, and assume that every edge in E is partitioned into a red end segment and a blue end segment, such that: (1) no two end segments of the same color cross; (2) every pair of end segments intersects at most once; and (3) no blue end segment is crossed by more than k red end segments that share a vertex. Assume further that G is drawn so that the number of edge crossings is minimized subject to the conditions (1)- (3). We show that G has O(kn) edges.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Graphs That Admit Polyline Drawings with Few Crossing Angles

Ackerman¹,

Fulek²,

Tóth³

2012

SIAM J. Discrete Math.

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“…Consequently research in that direction was intensified. In particular right angle crossing drawings (or short RAC drawings) were studied [5,11], and NP-hardness of the decision version for right angles was proven [2].…”

Section: Introductionmentioning

confidence: 99%

Graphs with Large Total Angular Resolution

Aichholzer

Korman

Okamoto

et al. 2019

Lecture Notes in Computer Science

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0000−0002−2364−0583] , Matias Korman 2 , Yoshio Okamoto 3[0000−0002−9826−7074] , Irene Parada 1[0000−0003−3147−0083] , Daniel Perz 1[0000−0002−6557−2355] , André van Renssen 4[0000−0002−9294−9947] , and Birgit Vogtenhuber 1[0000−0002−7166−4467]Abstract. The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with n vertices and a total angular resolution greater than 60 • is bounded by 2n − 6. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least 60 • is NP-hard.

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The Straight-Line RAC Drawing Problem Is NP-Hard

Cited by 26 publications

References 16 publications

Recognizing Optimal 1-Planar Graphs in Linear Time

Recognizing Optimal 1-Planar Graphs in Linear Time

Graphs That Admit Polyline Drawings with Few Crossing Angles

Graphs with Large Total Angular Resolution

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