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### On point-linear arboricity of planar graphs

Jianfang Wang
<span title="">1988</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Introduction Chartrand [l] studied the point-arboricity of planar graphs and showed that the point-arboricity of a planar graph does not exceed 3. Now we will discuss the point-linear arboricity.  ...  It can be proved that if G is a planar graph with JVJ = n s 8 then pO(G) < 2. However, there exists a planar graph with n = 9, its point-linear arboricity is larger than 2.  ...
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### Planar Projections of Graphs [chapter]

N. R. Aravind, Udit Maniyar
<span title="">2020</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;
Finally, we obtain bounds in terms of parameters such as geometric thickness and linear arboricity.  ...  We show that the complete graph on n vertices has a representation in ⌈√(n/2)+1 ⌉ planes.  ...  The linear arboricity of a graph G is the minimum number of linear forests into which the edges of G can be decomposed. We have the following. Proposition 1.  ...
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### Arboricity and bipartite subgraph listing algorithms

David Eppstein
<span title="">1994</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5xfy4fcdxvcvdiybp6iqbffqc4" style="color: black;">Information Processing Letters</a> </i> &nbsp;
In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs.  ...  The arboricity bound is necessary: for any constant k and any n there exists an n-vertex graph with O(n) edges and (n/ log n) k maximal complete bipartite subgraphs K k, .  ...  A decomposition of a planar graph into three forests can be found in linear time [7, 18] .  ...
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### Page 469 of Mathematical Reviews Vol. 40, Issue 3 [page]

<span title="">1970</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Circuit Theory (Ann Arbor, Mich., 1964), pp. 67-75, Univ. of Michigan Press, Ann Arbor, Mich., 1964] that an arbitrary graph is planar if and only if a suitable set of pseudo-Hamiltonian graphs, derived  ...  to identify planar sub- graphs of a nonplanar graph.  ...
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### Bounded diameter arboricity

Martin Merker, Luke Postle
<span title="2018-11-04">2018</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
We use these results to characterize the bounded diameter arboricity for planar graphs of girth at least g for all g = 5.  ...  Moreover, we answer a question by Mütze and Peter leading to an improved lower bound on the density of globally sparse Ramsey graphs.  ...  The first variant, introduced by Harary [10] in 1970 is linear arboricity, denoted Υ L (G), wherein the graph is decomposed into linear forests, where a linear forest is the disjoint union of paths.  ...
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### Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way

Matthew Dickerson, David Eppstein, Michael T. Goodrich, Jeremy Y. Meng
<span title="">2005</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;
ones) to test if a non-planar graph can be drawn efficiently in a confluent way.  ...  Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way.  ...  Acknowledgments We would like to thank Jie Ren and André van der Hoek for supplying us with several examples of graphs used in software visualization.  ...
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20051231080829/http://www.cs.brown.edu:80/publications/jgaa/accepted/2005/Dickerson+2005.9.1.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/21/44/21443144d398fec6e34435113edddb80147ebebf.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7155/jgaa.00099"> <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>

### Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way [chapter]

Matthew Dickerson, David Eppstein, Michael T. Goodrich, Jeremy Yu Meng
<span title="">2004</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;
ones) to test if a non-planar graph can be drawn efficiently in a confluent way.  ...  Our approach is to use a technique we call confluent drawing for visualizing non-planar graphs in a planar way.  ...  Acknowledgments We would like to thank Jie Ren and André van der Hoek for supplying us with several examples of graphs used in software visualization.  ...
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### Vertex and Tree Arboricities of Graphs

Gerard J. Chang, Chiuyuan Chen, Yaping Chen
<span title="">2004</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tsm6lnbvf5ahzjp7czojbjig5i" style="color: black;">Journal of combinatorial optimization</a> </i> &nbsp;
This paper studies the vertex and the tree arboricities on various classes of graphs for exact values, algorithms, bounds, hamiltonicity and NP-completeness.  ...  This paper studies the following variations of arboricity of graphs.  ...  Typical examples are linear arboricity, linear k-arboricity and star arboricity, whose definitions are the same as arboricity except that each subset of E(G) induces a graph whose components are paths,  ...
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### Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way [article]

Matthew Dickerson and David Eppstein and Michael T. Goodrich and Jeremy Meng
<span title="2002-12-17">2002</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Producing such confluent diagrams automatically from a graph with many crossings is quite challenging, however, so we offer two heuristic algorithms to test if a non-planar graph can be drawn efficiently  ...  In addition, we identify several large classes of graphs that can be completely categorized as being either confluently drawable or confluently non-drawable.  ...  Acknowledgments We would like to thank Jie Ren and André van der Hoek for supplying us with several examples of graphs used in software visualization.  ...
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### Page 20 of Mathematical Reviews Vol. 41, Issue 1 [page]

<span title="">1971</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The author states a similar “Pésa- type” result for n-connectedness and proves a generaliza- tion of Pésa’s theorem (case k=0): Let G be a graph on p points (vertices) and let 0<k<p-—2.  ...  Extremal nonseparable graphs of diameter 2. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968), pp. 111-118. Academic Press, New York, 1969.  ...
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### Three ways to cover a graph [article]

Kolja Knauer, Torsten Ueckerdt
<span title="2015-10-02">2015</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The classical graph parameters that result this way are interval number, track number, linear arboricity, star arboricity, and caterpillar arboricity.  ...  We provide new bounds on each covering number with respect to the following covering classes: linear forests, star forests, caterpillar forests, and interval graphs.  ...  West for useful comments improving the presentation of the paper.  ...
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### Adjacency queries in dynamic sparse graphs

Łukasz Kowalik
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5xfy4fcdxvcvdiybp6iqbffqc4" style="color: black;">Information Processing Letters</a> </i> &nbsp;
We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like for example planar graphs.  ...  This last result holds even for graphs of arboricity bounded by O(log k n), for some constant k.  ...  Intuitively, graphs of bounded arboricity are uniformly sparse, i.e., they do not have dense subgraphs. (In particular, planar graphs have arboricity 3.)  ...
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### Drawing Graphs on Few Lines and Few Planes [article]

Steven Chaplick, Krzysztof Fleszar, Fabian Lipp, Alexander Ravsky, Oleg Verbitsky, Alexander Wolff
<span title="2016-09-01">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.  ...  We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing  ...  Graph Algorithms Appl. 19(1), 393-412 (2015) 41. Wang, J.: On point-linear arboricity of planar graphs.  ...
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### Page 3176 of Mathematical Reviews Vol. , Issue 90F [page]

<span title="">1990</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
On the other hand, there are planar graphs with st(G) = 5.  ...  901:05097 90f:05097 05C50 Shearer, James B. (1-IBM) On the distribution of the maximum eigenvalue of graphs. Linear Algebra Appl. 114/115 (1989), 17-20.  ...
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### On graph thickness, geometric thickness, and separator theorems

Christian A. Duncan
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/is2k7576zbherpepqy3y7zkzuy" style="color: black;">Computational geometry</a> </i> &nbsp;
We investigate the relationship between geometric thickness, thickness, outerthickness, and arboricity of graphs.  ...  The technique used can be extended to other classes of graphs so long as a separator theorem exists.  ...  Acknowledgements We would like to thank David Eppstein and David Wood for several helpful discussions on this topic leading to the results in this paper and to the various referees for their helpful comments  ...
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