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### Homothetic triangle representations of planar graphs [article]

Daniel Gonçalves, Benjamin Lévêque, Alexandre Pinlou
<span title="2019-08-30">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that every planar graph is the intersection graph of homothetic triangles in the plane.  ...  In the present article, we focus on homothetic triangles. It has been shown that many planar graphs admit a contact representation with homothetic triangles.  ...  The following section is devoted to the proof of Theorem 2. Intersection representations with homothetic triangles It is well known that simple contact representations produce planar graphs.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1908.11749v1">arXiv:1908.11749v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cfb5losckjcmtfam3rcyh45cny">fatcat:cfb5losckjcmtfam3rcyh45cny</a> </span>
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### Contact representations of planar graphs with cubes

Stefan Felsner, Mathew C. Francis
<span title="">2011</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zqbrzpfj5rc3pcxpllbrit3n4q" style="color: black;">Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG &#39;11</a> </i> &nbsp;
We prove that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planar graph and two cubes have  ...  This result is a strengthening of a result by Thomassen which states that every planar graph has such a representation using axis-parallel boxes.  ...  The arrows indicate where the top corners of triangles touch. Figure 3 : 3 Figure 3: A homothetic triangle contact representation of a planar graph.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1998196.1998250">doi:10.1145/1998196.1998250</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/compgeom/FelsnerF11.html">dblp:conf/compgeom/FelsnerF11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/aauhnalzondqllllwsss7fgfza">fatcat:aauhnalzondqllllwsss7fgfza</a> </span>
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### Equilateral L-Contact Graphs [article]

Steven Chaplick and Stephen Kobourov and Torsten Ueckerdt
<span title="2013-03-06">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also study a slightly more restricted version of equilateral L-contact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs.  ...  We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm's monster packing theorem.  ...  Mchedlidze for many discussions about several variants of this problem.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1303.1279v1">arXiv:1303.1279v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ook2agsetbfcpk4y6m4fzzxrpy">fatcat:ook2agsetbfcpk4y6m4fzzxrpy</a> </span>
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### Equilateral L-Contact Graphs [chapter]

Steven Chaplick, Stephen G. Kobourov, Torsten Ueckerdt
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We also study a slightly more restricted version of equilateral L-contact systems and show that these are equivalent to homothetic triangle contact representations of maximally planar graphs.  ...  We believe that this new interpretation of the problem might allow for efficient algorithms to find homothetic triangle contact representations, that do not use Schramm's monster packing theorem. to point  ...  Finally, we consider a special class of equilateral L-representations, drawing connections to homothetic triangle contact representations of maximally planar graphs and contact representations with cubes  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-45043-3_13">doi:10.1007/978-3-642-45043-3_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oyabpqxt4vb6tlw5humctx2f4m">fatcat:oyabpqxt4vb6tlw5humctx2f4m</a> </span>
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### Triangle Contact Representations and Duality

Daniel Gonçalves, Benjamin Lévêque, Alexandre Pinlou
<span title="2012-02-16">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cu3ouk4pmzhsroy4l5k2c35xtu" style="color: black;">Discrete &amp; Computational Geometry</a> </i> &nbsp;
A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles  ...  A primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv  ...  So it is not possible to have an "exact" primal-dual contact representation by homothetic triangles for this graph.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00454-012-9400-1">doi:10.1007/s00454-012-9400-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xytvajmdrzhbrgk3prriaqhlp4">fatcat:xytvajmdrzhbrgk3prriaqhlp4</a> </span>
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### Triangle Contact Representations and Duality [chapter]

Daniel Gonçalves, Benjamin Lévêque, Alexandre Pinlou
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles  ...  A primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv  ...  So it is not possible to have an "exact" primal-dual contact representation by homothetic triangles for this graph.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-18469-7_24">doi:10.1007/978-3-642-18469-7_24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/buzgcccnxfft3ggnq2bwj3o7ca">fatcat:buzgcccnxfft3ggnq2bwj3o7ca</a> </span>
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### Cover Contact Graphs [chapter]

Nieves Atienza, Natalia de Castro, Carmen Cortés, M. Ángeles Garrido, Clara I. Grima, Gregorio Hernández, Alberto Márquez, Auxiliadora Moreno, Martin Nöllenburg, José Ramon Portillo, Pedro Reyes, Jesús Valenzuela (+2 others)
<span title="">2008</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Collins and Stephenson [CS03] give an efficient algorithm for numerically approximating the radii and locations of the disks of such a representation of a planar graph.  ...  Since Koebe there has been a lot of work in the graph-drawing community dedicated to the question which planar graphs can be represented as contact or intersection graphs of which geometric object.  ...  Introduction Koebe's theorem [Koe36, PA95] , a beautiful and classical result in graph theory, says that every planar graph can be represented as a coin graph, that is, a contact graph of disks in the  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-77537-9_18">doi:10.1007/978-3-540-77537-9_18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g4dd2kz7eve5nlmb3pj46hfbwi">fatcat:g4dd2kz7eve5nlmb3pj46hfbwi</a> </span>
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### Proportional Contact Representations of Planar Graphs

Md. Jawaherul Alam, Therese Biedl, Stefan Felsner, Michael Kaufmann, Stephen G. Kobourov
<span title="">2012</span> <i title="Journal of Graph Algorithms and Applications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4joumle7hffilk276fkubvhxcq" style="color: black;">Journal of Graph Algorithms and Applications</a> </i> &nbsp;
We describe algorithms for proportional contact representations with optimal polygonal complexity for general planar graphs and planar 2-segment graphs, which include maximal outer-planar graphs and partial  ...  We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by point-contacts or side-contacts between the corresponding polygons.  ...  A preliminary version of this paper was presented at Graph Drawing 2011 [2] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7155/jgaa.00276">doi:10.7155/jgaa.00276</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tz4ktdgfvja4nmipt4q5qfd22i">fatcat:tz4ktdgfvja4nmipt4q5qfd22i</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706171226/http://www.emis.de/journals/JGAA/accepted/2012/Alam+2012.16.3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/23/24/2324d34e77a60367282e0b6fb9f81a093fd5e808.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7155/jgaa.00276"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

### Proportional Contact Representations of Planar Graphs [chapter]

Muhammad Jawaherul Alam, Therese Biedl, Stefan Felsner, Michael Kaufmann, Stephen G. Kobourov
<span title="">2012</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We describe algorithms for proportional contact representations with optimal polygonal complexity for general planar graphs and planar 2-segment graphs, which include maximal outer-planar graphs and partial  ...  We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by point-contacts or side-contacts between the corresponding polygons.  ...  A preliminary version of this paper was presented at Graph Drawing 2011 [2] .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-25878-7_4">doi:10.1007/978-3-642-25878-7_4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/col4rf3b3na2fkym4uk5udhmge">fatcat:col4rf3b3na2fkym4uk5udhmge</a> </span>
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### Equiangular Polygon Contact Representations [chapter]

Stefan Felsner, Hendrik Schrezenmaier, Raphael Steiner
<span title="">2018</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Planar graphs are known to have contact representations of various types. The most prominent example is Koebe's 'kissing coins theorem'.  ...  Monster Packing implies the existence of contact representations of planar triangulations where each vertex v is represented by a homothetic copy of some smooth strictly-convex prototype P v .  ...  We want to thank Manfred Scheucher for supporting us with the implementation of the heuristic.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-030-00256-5_17">doi:10.1007/978-3-030-00256-5_17</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bdt44p32y5cs5is4oba6apdebq">fatcat:bdt44p32y5cs5is4oba6apdebq</a> </span>
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### Planar graphs as L-intersection or L-contact graphs [article]

Daniel Gonçalves, Claire Pennarun
<span title="2017-07-28">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that planar graphs are L-intersection graphs, and that triangle-free planar graphs are L, |, ---contact graphs.  ...  A subfamily of these graphs are L, |, ---contact graphs which are the contact graphs of axis parallel L, |, and -- shapes in the plane.  ...  Introduction The representation of graphs by contact or intersection of predefined shapes in the plane is a broad subject of research since the work of Koebe on the representation of planar graphs by contacts  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1707.08833v2">arXiv:1707.08833v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bpyuicbnkval5hjtgnbsffeevu">fatcat:bpyuicbnkval5hjtgnbsffeevu</a> </span>
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### Planar Graphs as L-intersection or L-contact graphs [chapter]

Daniel Gonçalves, Lucas Isenmann, Claire Pennarun
<span title="">2018</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ja7dpjgz2nhjlovlil3ij5m7ei" style="color: black;">Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms</a> </i> &nbsp;
We show that planar graphs are -intersection graphs, and that triangle-free planar graphs are { , |, −}-contact graphs.  ...  A subfamily of these graphs are { , |, −}-contact graphs which are the contact graphs of axis parallel , |, and − shapes in the plane.  ...  Introduction The representation of graphs by contact or intersection of predefined shapes in the plane is a broad subject of research since the work of Koebe on the representation of planar graphs by contacts  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/1.9781611975031.12">doi:10.1137/1.9781611975031.12</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/soda/GoncalvesIP18.html">dblp:conf/soda/GoncalvesIP18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qaot7xh2avedbhk5f7luujfsiq">fatcat:qaot7xh2avedbhk5f7luujfsiq</a> </span>
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### Axis-Aligned Square Contact Representations [article]

Andrew Nathenson
<span title="2021-07-25">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We introduce a new class 𝒢 of bipartite plane graphs and prove that each graph in 𝒢 admits a proper square contact representation.  ...  For every graph G∈𝒢, we construct a proper square contact representation. The key parameter of the recursive construction is the aspect ratio of the rectangle bounded by the four outer squares.  ...  Let G be a family of plane bipartite graphs defined recursively as follows.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.08719v2">arXiv:2103.08719v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vgrsdlxbtrag7htw42bp7ujvdi">fatcat:vgrsdlxbtrag7htw42bp7ujvdi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210730162025/https://arxiv.org/pdf/2103.08719v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e1/bf/e1bf244d459912ddeb89b76e3933bf72d71c6213.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.08719v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Pentagon Contact Representations

Stefan Felsner, Hendrik Schrezenmaier, Raphael Steiner
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fhi2xwpnh5gmlgof2idwu5wlgq" style="color: black;">Electronic Notes in Discrete Mathematics</a> </i> &nbsp;
Representations of planar triangulations as contact graphs of a set of internally disjoint homothetic triangles or respectively of a set of internally disjoint homothetic squares have received quite some  ...  In this paper we investigate representations of planar triangulations as contact graphs of a set of internally disjoint homothetic pentagons.  ...  In this paper we investigate representations of planar triangulations as contact graphs of a set of internally disjoint homothetic pentagons.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2017.06.069">doi:10.1016/j.endm.2017.06.069</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ng7l5mjmpzfvxfryrcdbfzycfe">fatcat:ng7l5mjmpzfvxfryrcdbfzycfe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200309173646/http://page.math.tu-berlin.de/~felsner/Paper/fcf-long.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/16/8e/168ee7820ca059a5f6577e45b65d683afa7204d2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.endm.2017.06.069"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### Linear-Time Algorithms for Hole-free Rectilinear Proportional Contact Graph Representations

M. Jawaherul Alam, Therese Biedl, Stefan Felsner, Andreas Gerasch, Michael Kaufmann, Stephen G. Kobourov
<span title="2013-03-16">2013</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qhi3z76be5c5xeihac5cyiid3m" style="color: black;">Algorithmica</a> </i> &nbsp;
There is an algorithm that gives a hole-free proportional contact representation of a maximal planar graph with 12-sided rectilinear polygons in O(n log n) time.  ...  We improve this result by giving a linear-time algorithm that produces a hole-free proportional contact representation of a maximal planar graph with a 10-sided rectilinear polygons.  ...  [1] show that planar partial 3-trees and some seriesparallel graphs have contact representations with homothetic triangles. Recently, Gonçalves et al.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00453-013-9764-5">doi:10.1007/s00453-013-9764-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jgc76lhl2nfutdu6fwpm5x5c6i">fatcat:jgc76lhl2nfutdu6fwpm5x5c6i</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20120124191755/http://page.math.tu-berlin.de/~felsner/Paper/lintime-contact.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/bf/2a/bf2a32c1fa55e807f51acd8e65279ab203027bd0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00453-013-9764-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>
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