4 Search Results for "Klemz, Boris"

Document
DOI: 10.4230/LIPIcs.ESA.2021.57
Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing

Authors: Boris Klemz

Published in: LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

Abstract
A hierarchical plane st-graph H can be thought of as a combinatorial description of a planar drawing Γ of a 2-connected graph G in which each edge is a y-monotone curve and each face encloses a y-monotone region (that is, a region whose intersection with any horizontal line is a line segment, a point, or empty). A drawing Γ' of H is a drawing of G such that each horizontal line intersects the same left-to-right order of edges and vertices in Γ and Γ', that is, the underlying hierarchical plane st-graph of both drawings is H. A straight-line planar drawing of a graph is convex if the boundary of each face is realized as a convex polygon. We study the computation of convex drawings of hierarchical plane st-graphs such that the outer face is realized as a prescribed polygon. Chrobak, Goodrich, and Tamassia [SoCG'96] and, independently, Kleist et al. [CGTA'19] described an idea to solve this problem in O(n^{1.1865}) time, where n is the number of vertices of the graph. Also independently, Hong and Nagamochi [J. Discrete Algorithms'10] described a completely different approach, which can be executed in O(n²) time. In this paper, we present an optimal O(n) time algorithm to solve the above problem, thereby improving the previous results by Chrobak, Goodrich, and Tamassia, Kleist et al., and by Hong and Nagamochi. Our result has applications in graph morphing. A planar morph is a continuous deformation of a graph drawing that preserves straight-line planarity. We show that our algorithm can be used as a drop-in replacement to speed up a procedure by Alamdari et al. [SICOMP'17] to morph between any two given straight-line planar drawings of the same plane graph. The running time improves from O(n^{2.1865}) to O(n²log n). To obtain our results, we devise a new strategy for computing so-called archfree paths in hierarchical plane st-graphs, which might be of independent interest.
Cite as

Boris Klemz. Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{klemz:LIPIcs.ESA.2021.57,
  author =	{Klemz, Boris},
  title =	{{Convex Drawings of Hierarchical Graphs in Linear Time, with Applications to Planar Graph Morphing}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{57:1--57:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.57},
  URN =		{urn:nbn:de:0030-drops-146381},
  doi =		{10.4230/LIPIcs.ESA.2021.57},
  annote =	{Keywords: convex drawing, hierarchical graph, graph drawing, computational geometry, planarity, planar graph, morphing, convexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.11
Adjacency Graphs of Polyhedral Surfaces

Authors: Elena Arseneva, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, and Alexander Wolff

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We study whether a given graph can be realized as an adjacency graph of the polygonal cells of a polyhedral surface in ℝ³. We show that every graph is realizable as a polyhedral surface with arbitrary polygonal cells, and that this is not true if we require the cells to be convex. In particular, if the given graph contains K_5, K_{5,81}, or any nonplanar 3-tree as a subgraph, no such realization exists. On the other hand, all planar graphs, K_{4,4}, and K_{3,5} can be realized with convex cells. The same holds for any subdivision of any graph where each edge is subdivided at least once, and, by a result from McMullen et al. (1983), for any hypercube. Our results have implications on the maximum density of graphs describing polyhedral surfaces with convex cells: The realizability of hypercubes shows that the maximum number of edges over all realizable n-vertex graphs is in Ω(n log n). From the non-realizability of K_{5,81}, we obtain that any realizable n-vertex graph has 𝒪(n^{9/5}) edges. As such, these graphs can be considerably denser than planar graphs, but not arbitrarily dense.
Cite as

Elena Arseneva, Linda Kleist, Boris Klemz, Maarten Löffler, André Schulz, Birgit Vogtenhuber, and Alexander Wolff. Adjacency Graphs of Polyhedral Surfaces. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arseneva_et_al:LIPIcs.SoCG.2021.11,
  author =	{Arseneva, Elena and Kleist, Linda and Klemz, Boris and L\"{o}ffler, Maarten and Schulz, Andr\'{e} and Vogtenhuber, Birgit and Wolff, Alexander},
  title =	{{Adjacency Graphs of Polyhedral Surfaces}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{11:1--11:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.11},
  URN =		{urn:nbn:de:0030-drops-138107},
  doi =		{10.4230/LIPIcs.SoCG.2021.11},
  annote =	{Keywords: polyhedral complexes, realizability, contact representation}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.58
Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian

Authors: Michael Hoffmann and Boris Klemz

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
We show that every triconnected planar graph of maximum degree five is subhamiltonian planar. A graph is subhamiltonian planar if it is a subgraph of a Hamiltonian planar graph or, equivalently, if it admits a 2-page book embedding. In fact, our result is stronger because we only require vertices of a separating triangle to have degree at most five, all other vertices may have arbitrary degree. This degree bound is tight: We describe a family of triconnected planar graphs that are not subhamiltonian planar and where every vertex of a separating triangle has degree at most six. Our results improve earlier work by Heath and by Bauernöppel and, independently, Bekos, Gronemann, and Raftopoulou, who showed that planar graphs of maximum degree three and four, respectively, are subhamiltonian planar. The proof is constructive and yields a quadratic time algorithm to obtain a subhamiltonian plane cycle for a given graph. As one of our main tools, which might be of independent interest, we devise an algorithm that, in a given 3-connected plane graph satisfying the above degree bounds, collapses each maximal separating triangle into a single edge such that the resulting graph is biconnected, contains no separating triangle, and no separation pair whose vertices are adjacent.
Cite as

Michael Hoffmann and Boris Klemz. Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{hoffmann_et_al:LIPIcs.ESA.2019.58,
  author =	{Hoffmann, Michael and Klemz, Boris},
  title =	{{Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{58:1--58:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.58},
  URN =		{urn:nbn:de:0030-drops-111797},
  doi =		{10.4230/LIPIcs.ESA.2019.58},
  annote =	{Keywords: Graph drawing, book embedding, Hamiltonian graph, planar graph, bounded degree graph, graph augmentation, computational geometry, SPQR decomposition}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.37
Strongly Monotone Drawings of Planar Graphs

Authors: Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, and Manfred Scheucher

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract
A straight-line drawing of a graph is a monotone drawing if for each pair of vertices there is a path which is monotonically increasing in some direction, and it is called a strongly monotone drawing if the direction of monotonicity is given by the direction of the line segment connecting the two vertices. We present algorithms to compute crossing-free strongly monotone drawings for some classes of planar graphs; namely, 3-connected planar graphs, outerplanar graphs, and 2-trees. The drawings of 3-connected planar graphs are based on primal-dual circle packings. Our drawings of outerplanar graphs depend on a new algorithm that constructs strongly monotone drawings of trees which are also convex. For irreducible trees, these drawings are strictly convex.
Cite as

Stefan Felsner, Alexander Igamberdiev, Philipp Kindermann, Boris Klemz, Tamara Mchedlidze, and Manfred Scheucher. Strongly Monotone Drawings of Planar Graphs. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{felsner_et_al:LIPIcs.SoCG.2016.37,
  author =	{Felsner, Stefan and Igamberdiev, Alexander and Kindermann, Philipp and Klemz, Boris and Mchedlidze, Tamara and Scheucher, Manfred},
  title =	{{Strongly Monotone Drawings of Planar Graphs}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.37},
  URN =		{urn:nbn:de:0030-drops-59292},
  doi =		{10.4230/LIPIcs.SoCG.2016.37},
  annote =	{Keywords: graph drawing, planar graphs, strongly monotone, strictly convex, primal-dual circle packing}
}
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