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A Note on Minimum-Segment Drawings of Planar Graphs

Stephane Durocher, Debajyoti Mondal, Rahnuma Islam Nishat, Sue Whitesides
2013 Journal of Graph Algorithms and Applications  
A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment.  ...  Finally, we investigate a worst-case lower bound on the number of segments required by straight-line drawings of arbitrary spanning trees of a given planar graph.  ...  A natural question is: what is the time complexity of computing a minimum-segment drawing of a planar graph [2] ? Dujmović et al.  ... 
doi:10.7155/jgaa.00295 fatcat:mcfrjub535bmhiho7s6f6lxnaa

Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs [chapter]

Xuemin Lin, Peter Eades
1999 Lecture Notes in Computer Science  
Firstly, we provide a quadratic algorithm that minimizes the drawing area with respect to a xed planar embedding.  ...  In this paper, we shall investigate the computational complexity of constructing minimum area grid visibility representations of hierarchically planar graphs.  ...  Moreover, an e cient grid visibility representation algorithm was presented that can achieve the minimum drawing area for a hierarchically planar graph with only one \source", only one \sink", and a xed  ... 
doi:10.1007/3-540-48686-0_9 fatcat:u3mbk5b4ejegpabyxbttnnhgmi

The Complexity of Drawing a Graph in a Polygonal Region [article]

Anna Lubiw and Tillmann Miltzow and Debajyoti Mondal
2018 arXiv   pre-print
Our result is one of the first showing that a problem of drawing planar graphs with straight-line edges is hard for the existential theory of the reals.  ...  of the region, place the remaining vertices to create a planar straight-line drawing of the graph inside the region.  ...  We would like to thank Günter Rote, who discussed with the second author the turn gadget in the context of the Art Gallery Problem. The Complexity of Drawing a Graph in a Polygonal Region  ... 
arXiv:1802.06699v3 fatcat:gkzwt5a2jrffpg5dzroymbacv4

Capturing Lombardi Flow in Orthogonal Drawings by Minimizing the Number of Segments [article]

Md. Jawaherul Alam, Michael Dillencourt, Michael T. Goodrich
2016 arXiv   pre-print
We study two problems on orthogonal drawings of planar graphs, one that minimizes the total number of line segments and another that minimizes the number of line segments that cover all the vertices.  ...  For both graph classes, we give polynomial-time algorithms for upward orthogonal drawings with the minimum number of segments covering the vertices.  ...  This article reports on work supported by the Defense Advanced Research Projects Agency under agreement no. AFRL FA8750-15-2-0092.  ... 
arXiv:1608.03943v1 fatcat:veivfxkabremnkur4ii556cewm

Towards area requirements for drawing hierarchically planar graphs

Xuemin Lin, Peter Eades
2003 Theoretical Computer Science  
An application of the existing results from upward drawing can guarantee a quadric drawing area for grid visibility representation but does not necessarily guarantee the minimum drawing area.  ...  Motivated by this, we will present a new grid visibility drawing algorithm which is e cient and guarantees the minimum drawing area with respect to a given topological embedding.  ...  Acknowledgements The authors are very grateful to the anonymous referees for a careful review and many valuable comments for improving the presentation of the paper.  ... 
doi:10.1016/s0304-3975(02)00031-2 fatcat:llocylvu55hl5amntzu35n7lrq

Minimum Length Embedding of Planar Graphs at Fixed Vertex Locations [chapter]

Timothy M. Chan, Hella-Franziska Hoffmann, Stephen Kiazyk, Anna Lubiw
2013 Lecture Notes in Computer Science  
We consider the problem of finding a planar embedding of a graph at fixed vertex locations that minimizes the total edge length. The problem is known to be NP-hard.  ...  We give polynomial time algorithms achieving an O( √ n log n) approximation for paths and matchings, and an O(n) approximation for general graphs.  ...  We learned about Sam Loyd's disjoint paths puzzle (which is not original to him) from Marcus Schaefer who has studied the history of such planarity puzzles.  ... 
doi:10.1007/978-3-319-03841-4_33 fatcat:gj6aw4zwybf2hjgix2jcctgohu

On the Minimum Cut of Planarizations

Markus Chimani, Carsten Gutwenger, Petra Mutzel
2007 Electronic Notes in Discrete Mathematics  
Every drawing of a non-planar graph G in the plane induces a planarization, i.e., a planar graph obtained by replacing edge crossings with dummy vertices.  ...  On the other hand, we prove that every crossing minimal planarization can be efficiently transformed into another crossing minimal planarization that preserves the capacity of a minimum st-cut in G.  ...  In this paper, we deal with graph theoretic properties of the planar graph induced by a drawing of G.  ... 
doi:10.1016/j.endm.2007.01.036 fatcat:cxsebbtjlne2rkhnacydltmizi

Area requirements for drawing hierarchically planar graphs [chapter]

Xuemin Lin, Peter Eades
1997 Lecture Notes in Computer Science  
Two drawing standards will be discussed: 1) each vertex is represented by a point and 2) grid visibifity representation (that is, a line segment is allowed to represent a vertex).  ...  Applications of some existing algorithms from upward drawing can guarantee the quadratic drawing area for grid visibility representation but do not necessarily guarantee the minimum drawing area.  ...  Respecting a planar embedding EH of a hierarchically planar graph H, the grid visibility representation of H produced by the algorithm GVP has the minimum drawing area.  ... 
doi:10.1007/3-540-63938-1_64 fatcat:vqxhepwm2bec5p2vamstciax4u

Obstacle Numbers of Planar Graphs [article]

John Gimbel, Patrice Ossona de Mendez, Pavel Valtr
2017 arXiv   pre-print
If G is planar, we define the planar obstacle number of G by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of G).  ...  graphs of straight line segments in two directions) of order at least 3 is 1.  ...  Acknowledgments We thank Vít Jelínek for ideas leading to a simplification of our proof. We also thank Daniel Král ' and Roman Nedela for inspiring comments on our research.  ... 
arXiv:1706.06992v3 fatcat:ucgmzezbazd6rhqwfborctcjtq

Drawing Outer-Planar Graphs in O(n log n )Area [chapter]

Therese Biedl
2002 Lecture Notes in Computer Science  
In this paper, we study drawings of outer-planar graphs in various models.  ...  The question of straight-line grid-drawings of outerplanar graphs in o(n 2 ) area remains open.  ...  There exists an outer-planar graph G such that any poly-line drawing Γ of G with all vertices on the boundary of the minimum enclosing rectangle of the graph has area Ω(n 2 ). Proof.  ... 
doi:10.1007/3-540-36151-0_6 fatcat:n67ueprnbvbn3ef4h3btbir5va

Orthogonal drawings of graphs with vertex and edge labels

Carla Binucci, Walter Didimo, Giuseppe Liotta, Maddalena Nonato
2005 Computational geometry  
This paper studies the problem of computing orthogonal drawings of graphs with labels on vertices and edges.  ...  We present MILP models to compute optimal drawings with respect to the first three goals and an exact algorithm that is based on these models to compute a labeled drawing of minimum area.  ...  Orthogonal drawings A planar orthogonal drawing Γ of a planar graph G is a planar drawing of G such that each edge e is mapped to a sequence of horizontal and vertical segments: a left or a right turn  ... 
doi:10.1016/j.comgeo.2005.02.001 fatcat:whimtnqjq5f4pcp6fx6l7f2o4q

Slanted Orthogonal Drawings [chapter]

Michael A. Bekos, Michael Kaufmann, Robert Krug, Stefan Näher, Vincenzo Roselli
2013 Lecture Notes in Computer Science  
We introduce a new model that we call slanted orthogonal graph drawing.  ...  While in traditional orthogonal drawings each edge is made of axis-aligned line-segments, in slanted orthogonal drawings intermediate diagonal segments on the edges are also permitted, which allows for  ...  Note that the general problem of determining a planar embedding with the minimum number of bends is NP-hard [5] , which is also the case for slog drawings.  ... 
doi:10.1007/978-3-319-03841-4_37 fatcat:5stn4riwybebxo2zyzojznwgxe

Really Straight Graph Drawings [chapter]

Vida Dujmović, Matthew Suderman, David R. Wood
2005 Lecture Notes in Computer Science  
We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes, and that every cubic 3-connected plane graph has a plane drawing with three  ...  We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees.  ...  Acknowledgements Thanks to all of the participants of the Bellairs workshop for creating a stimulating working environment. Special thanks to Mike Fellows for suggesting the problem.  ... 
doi:10.1007/978-3-540-31843-9_14 fatcat:fwradttc6ba7jip6ftrdahsdwq

Smooth Orthogonal Layouts [chapter]

Michael A. Bekos, Michael Kaufmann, Stephen G. Kobourov, Antonios Symvonis
2013 Lecture Notes in Computer Science  
We study the problem of creating smooth orthogonal layouts for planar graphs.  ...  Using the Kandinsky model for removing the degree restriction, we show that any planar graph has a smooth complexity-2 layout. The work of M.A.  ...  Let G be a 4-planar graph that admits an orthogonal drawing H with minimum number of bends and edge complexity k.  ... 
doi:10.1007/978-3-642-36763-2_14 fatcat:5zg4c4omgrekrmuocaklaaqxha

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 seminar brought together 29 researchers in the areas of graph theory, combinatorics, computational geometry, and graph drawing.  ...  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  ...  A note on minimum-segment drawings of planar graphs. J. Graph Algorithms Appl., 17:301-328, 2013. 14 David Eppstein, Marc van Kreveld, Elena Mumford, and Bettina Speckmann.  ... 
doi:10.4230/dagrep.6.11.35 dblp:journals/dagstuhl-reports/Hong0KP16 fatcat:gmft5jywxbapza52lhj4o23ngq
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