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

Atsushi Kaneko, Hiroshi Maehara, Mamoru Watanabe
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.  ... 
doi:10.1006/jctb.1998.1869 fatcat:ecchz3apwzgjxf4uoanh7xsoqu

Acute triangles in 4-connected maximal plane graphs

Ken-ichi Kawarabayashi, Atsuhiro Nakamoto, Yoshiaki Oda, Mamoru Watanabe
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.  ... 
doi:10.1016/j.disc.2004.09.008 fatcat:tgtsxiwwmvfwhbrwmfbgkxmyge

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

Kenji Koyama, Atsuhiro Nakamoto
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.  ... 
doi:10.1016/j.disc.2005.01.009 fatcat:vkbdwi4zzjh77benueppjttbeu

Drawing Partially Embedded and Simultaneously Planar Graphs

Timothy M. Chan, Fabrizio Frati, Carsten Gutwenger, Anna Lubiw, Petra Mutzel, Marcus Schaefer
2015 Journal of Graph Algorithms and Applications  
We investigate 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.  ... 
doi:10.7155/jgaa.00375 fatcat:i7opm6avcngzrcx73dfekq4jpm

Drawing Partially Embedded and Simultaneously Planar Graphs [chapter]

Timothy M. Chan, Fabrizio Frati, Carsten Gutwenger, Anna Lubiw, Petra Mutzel, Marcus Schaefer
2014 Lecture Notes in Computer Science  
We investigate 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.  ... 
doi:10.1007/978-3-662-45803-7_3 fatcat:7kjeaklwazhy7ph3qlrh4k6a4m

Drawing graphs with right angle crossings

Walter Didimo, Peter Eades, Giuseppe Liotta
2011 Theoretical Computer Science  
Cognitive experiments show that humans can read 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.  ... 
doi:10.1016/j.tcs.2011.05.025 fatcat:aapalbfrorfmvcdiacvhm4ss64

The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree

Vít Jelínek, Eva Jelínková, Jan Kratochvíl, Bernard Lidický, Marek Tesař, Tomáš Vyskočil
2012 Graphs and Combinatorics  
We study 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.  ... 
doi:10.1007/s00373-012-1157-z fatcat:azhsisihuzabbefsog27kvimo4

The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree [chapter]

Vít Jelínek, Eva Jelínková, Jan Kratochvíl, Bernard Lidický, Marek Tesař, Tomáš Vyskočil
2010 Lecture Notes in Computer Science  
We study 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.  ... 
doi:10.1007/978-3-642-11805-0_29 fatcat:uzxadd6a7rbspjzgtkiomktxca

The Straight-Line RAC Drawing Problem is NP-Hard

Evmorfia N. Argyriou, Michael A. Bekos, Antonios Symvonis
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).  ... 
doi:10.7155/jgaa.00274 fatcat:r6ybon2kb5b3vnpisgvspsag34

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

Evmorfia N. Argyriou, Michael A. Bekos, Antonios Symvonis
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).  ... 
doi:10.1007/978-3-642-18381-2_6 fatcat:n7oiljtaj5dazanry2r2vjan74

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

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, János Pach, Marc Herbstritt
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.  ... 
doi:10.4230/dagrep.6.11.35 dblp:journals/dagstuhl-reports/Hong0KP16 fatcat:gmft5jywxbapza52lhj4o23ngq

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

William Lenhart, Giuseppe Liotta, Debajyoti Mondal, Rahnuma Islam Nishat
2013 Lecture Notes in Computer Science  
As 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 .  ... 
doi:10.1007/978-3-319-03841-4_36 fatcat:aizqfijgnjfajgqz6b2irafuaa

Arrangements of orthogonal circles with many intersections [article]

Sarah Carmesin, André Schulz
2021 arXiv   pre-print
Based 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.  ... 
arXiv:2106.03557v2 fatcat:fwi4mwlf3ffgfpoceb6k46uxtm

Survey of two-dimensional acute triangulations

Carol T. Zamfirescu
2013 Discrete Mathematics  
upper bounds, and 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.  ... 
doi:10.1016/j.disc.2012.09.016 fatcat:5dr3jucig5b6rf5umlsfcdskrq

POLYNOMIAL-SIZE NONOBTUSE TRIANGULATION OF POLYGONS

MARSHALL BERN, DAVID EPPSTEIN
1992 International journal of computational geometry and applications  
We describe methods for triangulating polygonal regions 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  ... 
doi:10.1142/s0218195992000159 fatcat:j6lf7l7x45gbzj2nwexlykx7t4
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