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On the Number of Acute Triangles in a Straight-Line Embedding of a Maximal Planar Graph

1999
*
Journal of combinatorial theory. Series B (Print)
*

*In*this paper we show that any

*maximal*

*planar*

*graph*with m

*triangles*except

*the*unbounded face can be transformed into

*a*

*straight*-

*line*

*embedding*

*in*which at least WmÂ3X

*triangles*are

*acute*

*triangles*. ... Every

*planar*

*embedding*

*of*

*a*

*planar*

*graph*can be transformed into an

*embedding*

*in*which

*the*edges are

*straight*

*line*segments. ... INTRODUCTION

*A*

*planar*

*embedding*

*of*

*a*

*planar*

*graph*is an

*embedding*

*of*

*the*

*graph*

*in*

*the*plane with no crossing.

*A*

*straight*-

*line*

*embedding*is an

*embedding*

*in*which

*the*edges are

*straight*

*line*segments. ...

##
###
Acute triangles in 4-connected maximal plane graphs

2005
*
Discrete Mathematics
*

*In*this paper, we show that every 4-connected

*maximal*plane

*graph*with m finite faces other than

*the*octahedron can be drawn

*in*

*the*plane so that at least (m + 3)/2 faces are

*acute*

*triangles*. ... Acknowledgements

*The*authors would like to thank

*the*referees for their careful readings and appropriate comments and advice. ... Every

*planar*

*graph*has

*a*

*straight*-

*line*

*embedding*.

*In*this paper, we focus

*on*

*a*"good"

*straight*-

*line*

*embedding*

*of*

*a*plane triangulation. ...

##
###
Acute triangles in triangulations on the plane with minimum degree at least 4

2005
*
Discrete Mathematics
*

*In*this paper, we show that every

*maximal*plane

*graph*with minimum degree at least 4 and m finite faces other than an octahedron can be drawn

*in*

*the*plane so that at least (m + 3)/2 faces are

*acute*

*triangles*...

*A*

*planar*

*embedding*

*of*G is called

*a*

*straight*-

*line*

*embedding*if each edge

*of*G is

*a*

*straight*-

*line*segment

*in*

*the*

*planar*

*embedding*.

*The*following theorem was proved by Wagner [6] and Fáry [1] . ... Every

*planar*

*graph*has

*a*

*straight*-

*line*

*embedding*.

*A*plane

*graph*G is said to be

*a*triangulation if each face

*of*G is triangular. ...

##
###
Drawing Partially Embedded and Simultaneously Planar Graphs

2015
*
Journal of Graph Algorithms and Applications
*

We investigate

doi:10.7155/jgaa.00375
fatcat:i7opm6avcngzrcx73dfekq4jpm
*the*problem*of*constructing*planar*drawings with few bends for two related problems,*the*partially*embedded**graph*problem-to extend*a**straight*-*line**planar*drawing*of**a*subgraph to*a**planar*... drawing*of**the*whole*graph*-and*the*simultaneous*planarity*problem-to find*planar*drawings*of*two*graphs*that coincide*on*shared vertices and edges. ...*The*authors would like to thank*the*anonymous referees for their useful comments and suggestions. ...##
###
Drawing Partially Embedded and Simultaneously Planar Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

We investigate

doi:10.1007/978-3-662-45803-7_3
fatcat:7kjeaklwazhy7ph3qlrh4k6a4m
*the*problem*of*constructing*planar*drawings with few bends for two related problems,*the*partially*embedded**graph*problem-to extend*a**straight*-*line**planar*drawing*of**a*subgraph to*a**planar*... drawing*of**the*whole*graph*-and*the*simultaneous*planarity*problem-to find*planar*drawings*of*two*graphs*that coincide*on*shared vertices and edges. ...*The*authors would like to thank*the*anonymous referees for their useful comments and suggestions. ...##
###
Drawing graphs with right angle crossings

2011
*
Theoretical Computer Science
*

Cognitive experiments show that humans can read

doi:10.1016/j.tcs.2011.05.025
fatcat:aapalbfrorfmvcdiacvhm4ss64
*graph*drawings*in*which all edge crossings are at right angles equally well as they can read*planar*drawings; they also show that*the*readability*of**a*drawing ... We establish upper and lower bounds*on*these quantities by considering two classical*graph*drawing scenarios:*The**one*where*the*algorithm can choose*the*combinatorial*embedding**of**the*input*graph*and*the*... Acknowledgements We acknowledge*the*anonymous reviewers for their valuable comments. ...##
###
The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree

2012
*
Graphs and Combinatorics
*

We study

doi:10.1007/s00373-012-1157-z
fatcat:azhsisihuzabbefsog27kvimo4
*the**planar*slope*number*, i.e.,*the*minimum*number**of*distinct edge-slopes*in*such*a*drawing*of**a**planar**graph*with maximum degree Δ. ... It is known that every*planar**graph*has*a**planar**embedding*where edges are represented by non-crossing*straight*-*line*segments. ...*In*this paper, we examine*the*minimum*number**of*slopes*in**a**straight*-*line**embedding**of**a**planar**graph*. ...##
###
The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree
[chapter]

2010
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-642-11805-0_29
fatcat:uzxadd6a7rbspjzgtkiomktxca
*the**planar*slope*number*, i.e.,*the*minimum*number**of*distinct edge-slopes*in*such*a*drawing*of**a**planar**graph*with maximum degree ∆. ... It is known that every*planar**graph*has*a**planar**embedding*where edges are represented by non-crossing*straight*-*line*segments. ...*In*this paper, we examine*the*minimum*number**of*slopes*in**a**straight*-*line**embedding**of**a**planar**graph*. ...##
###
The Straight-Line RAC Drawing Problem is NP-Hard

2012
*
Journal of Graph Algorithms and Applications
*

*In*this paper, we focus

*on*

*straight*-

*line*RAC drawings and demonstrate an infinite class

*of*

*graphs*with unique RAC combinatorial

*embedding*. ... We employ members

*of*this class

*in*order to show that it is N P-hard to decide whether

*a*

*graph*admits

*a*

*straight*-

*line*RAC drawing. ... Note that it is not feasible

*a*non-

*planar*,

*straight*-

*line*RAC drawing

*of*

*a*quadrilateral to contain more than

*one*right-angle crossing (

*the*drawing

*of*Q forms two orthogonal

*triangles*). ...

##
###
The Straight-Line RAC Drawing Problem Is NP-Hard
[chapter]

2011
*
Lecture Notes in Computer Science
*

*In*this work, we demonstrate

*a*class

*of*

*graphs*with unique RAC combinatorial

*embedding*and we employ members

*of*this class

*in*order to show that it is NP-hard to decide whether

*a*

*graph*admits

*a*

*straight*-

*line*... Recent cognitive experiments have shown that

*the*negative impact

*of*an edge crossing

*on*

*the*human understanding

*of*

*a*

*graph*drawing, tends to be eliminated

*in*

*the*case where

*the*crossing angles are greater ... Note that it is not feasible

*a*non-

*planar*,

*straight*-

*line*RAC drawing

*of*

*a*quadrilateral to contain more than

*one*right-angle crossing (

*the*drawing

*of*Q forms two orthogonal

*triangles*). ...

##
###
Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

2017
*
Dagstuhl Reports
*

*The*common interest was

*in*

*the*exploration

*of*structural properties and

*the*development

*of*algorithms for so-called beyond-

*planar*

*graphs*, i.e., non-

*planar*

*graphs*with topological constraints such as specific ...

*The*seminar brought together 29 researchers

*in*

*the*areas

*of*

*graph*theory, combinatorics, computational geometry, and

*graph*drawing. ... This idea is captured by

*the*segment

*number*

*of*

*a*

*graph*, that is,

*the*smallest

*number*

*of*

*line*segments that together constitute

*a*

*straight*-

*line*drawing

*of*

*the*given

*graph*. ...

##
###
Planar and Plane Slope Number of Partial 2-Trees
[chapter]

2013
*
Lecture Notes in Computer Science
*

As

doi:10.1007/978-3-319-03841-4_36
fatcat:aizqfijgnjfajgqz6b2irafuaa
*a*byproduct*of*our techniques, we answer*a*long standing question by Garg and Tamassia about*the*angular resolution*of**the**planar**straight*-*line*drawings*of*series-parallel*graphs**of*bounded degree. ... We prove tight bounds (up to*a*small multiplicative or additive constant) for*the*plane and*the**planar*slope*numbers**of*partial 2-trees*of*bounded degree. ...*The*slope*number**of**a**straight*-*line*drawing Γ*of**a**planar**graph*G is*the**number**of*distinct slopes*of**the*edges*of*Γ . ...##
###
Arrangements of orthogonal circles with many intersections
[article]

2021
*
arXiv
*
pre-print

Based

arXiv:2106.03557v2
fatcat:fwi4mwlf3ffgfpoceb6k46uxtm
*on**the*lower bound we can also improve*the*bound for*the**number**of**triangles**in*arrangements*of*orthogonal circles to (3 + 5/9)n-O(√(n)). ... For*the*general case we prove that*the**maximal**number**of*edges*in*an intersection*graph**of*an arrangement*of*orthogonal circles lies*in*between 4n - O(√(n)) and (4+5/11)n, for n being*the**number**of*circles ...*One*direction*of**the*circle packing theorem is obvious,*a**planar**straight*-*line*drawing*of**the*contact*graph*can be derived by placing*the*vertices at*the*disk centers. ...##
###
Survey of two-dimensional acute triangulations

2013
*
Discrete Mathematics
*

upper bounds, and

doi:10.1016/j.disc.2012.09.016
fatcat:5dr3jucig5b6rf5umlsfcdskrq
*a*detour into dissections and*acute*triangulations*of**planar**graphs*. ...*The*corresponding*planar*problem was treated by Wagner [75] . See also [29, 70] .*A*triangulation is non-obtuse (*acute*) if all angles within its*triangles*are at most (strictly less than) π /2. ... West, Tudor Zamfirescu, and*the*anonymous referees for their many helpful suggestions and constructive criticism. ...##
###
POLYNOMIAL-SIZE NONOBTUSE TRIANGULATION OF POLYGONS

1992
*
International journal of computational geometry and applications
*

We describe methods for triangulating polygonal regions

doi:10.1142/s0218195992000159
fatcat:j6lf7l7x45gbzj2nwexlykx7t4
*of**the*plane so that no*triangle*has*a*large angle. ... We also show that any triangulation (without Steiner points)*of**a*simple polygon has*a*refinement with O(n 4 ) nonobtuse*triangles*. ... Acknowledgements We would like to thank Warren Smith for drawing our attention to this problem, Anna Lubiw for*the*term "path polygon", and David Dobkin, John Gilbert, and Mike Paterson for some helpful ...
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