ABSTRACT
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron with the topology of a sphere in which three mutually-perpendicular edges meet at each vertex. By analogy to Steinitz's theorem characterizing the graphs of convex polyhedra, we characterize the graphs of simple orthogonal polyhedra: they are exactly the 3-regular bipartite planar graphs in which the removal of any two vertices produces at most two connected components. We also characterize two subclasses of these polyhedra: corner polyhedra, which can be drawn by isometric projection in the plane with only one hidden vertex, and xyz polyhedra, in which each axis-parallel line through a vertex contains exactly one other vertex. Based on our characterizations we find efficient algorithms for constructing orthogonal polyhedra from their graphs
- A. A. Andersson and M. Thorup. Tight(er) worst-case bounds on dynamic searching and priority queues. Proc. 32nd ACM Symp. Theory of Computing (STOC 2000), pp. 335--342, 2000, doi:10.1145/335305.335344. Google ScholarDigital Library
- S. Aziza and T. Biedl. Hexagonal grid drawings: algorithms and lower bounds. Proc. 12th Int. Symp. Graph Drawing (GD 2004), 3383 edition, pp. 18--24. Springer-Verlag, Lecture Notes in Computer Science, 2005. Google ScholarDigital Library
- M. L. Balinski. On the graph structure of convex polyhedra in n-space. Pacific J. Math. 11(2):431--434, 1961, http://projecteuclid.org/euclid.pjm/1103037323.Google ScholarCross Ref
- D. W. Barnette. Conjecture 5. Recent Progress in Combinatorics, p. 343. Academic Press, 1969.Google Scholar
- V. Batagelj. An improved inductive definition of two restricted classes of triangulations of the plane. Combinatorics and Graph Theory, Warsaw, 1987, 25 edition, pp. 11--18. PWN, Banach Center Publ., 1989.Google Scholar
- S. Bhatt and S. Cosmodakis. The complexity of minimizing wire lengths in VLSI layouts. Inform. Proc. Lett. 25:263--267, 1987, doi:10.1016/0020-0190(87)90173-6. Google ScholarDigital Library
- T. Biedl, E. Demaine, M. Demaine, A. Lubiw, M. Overmars, J. O'Rourke, S. Robbins, and S. Whitesides. Unfolding some classes of orthogonal polyhedra. Proc. 10th Canadian Conference on Computational Geometry (CCCG'98), 1998, http://erikdemaine.org/papers/CCCG98b/.Google Scholar
- T. Biedl and B. Genc. When can a graph form an orthogonal polyhedron? Proc. 16th Canadian Conf. Computational Geometry (CCCG 2004), pp. 53--56, 2004, http://www.cccg.ca/proceedings/2004/15.pdf.Google Scholar
- T. Biedl and B. Genc. Cauchy's theorem for orthogonal polyhedra of genus 0. Proc. 17th European Symp. Algorithms (ESA 2009), 5757 edition, pp. 71--83. Springer-Verlag, Lecture Notes in Computer Science, 2009, doi:10.1007/978-3-642-04128-0_7.Google ScholarCross Ref
- J. Bokowski and A. Guedes de Oliveira. On the generation of oriented matroids. Discrete and Computational Geometry 24(2-3):197--208, 2000, doi:10.1007/s004540010027.Google ScholarCross Ref
- G. Brinkmann and B. D. McKay. Fast generation of planar graphs. Communications in Mathematical and in Computer Chemistry 58(2):323--357, 2007, http://cs.anu.edu.au/people/bdm/papers/plantri-full.pdf.Google Scholar
- M. Bruls, K. Huizing, and J. J. van Wijk. Squarified treemaps. Data Visualization 2000: Proc. Joint Eurographics and IEEE TCVG Symp. on Visualization, pp. 33--42. Springer-Verlag, 2000, http://www.win.tue.nl/~vanwijk/stm.pdf.Google Scholar
- A. L. Cauchy. Sur les polygones et polyèdres. J. École Polytechnique 19:87--98, 1813.Google Scholar
- N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM J. Comput. 14(1):210--223, 1985, doi:10.1137/0214017. Google ScholarDigital Library
- M. Chrobak and D. Eppstein. Planar orientations with low out-degree and compaction of adjacency matrices. Theoretical Computer Science 86(2):243--266, September 1991, doi:10.1016/0304-3975(91)90020-3. Google ScholarDigital Library
- H. Cohn, M. Larsen, and J. Propp. The shape of a typical boxed plane partition. New York J. Math. 4:137--165, 1998, http://arxiv.org/abs/math.CO/9801059arXiv:math.CO/9801059, http://nyjm.albany.edu:8000/j/1998/4_137.html.Google Scholar
- R. Connelly. A counterexample to the rigidity conjecture for polyhedra. Pub. Math. de l'IHÉS 47:333--338, 1977, doi:10.1007/BF02684342.Google ScholarCross Ref
- R. Connelly, I. Sabitov, and A. Walz. The bellows conjecture. Beitrage Algebra Geom. 38:1--10, 1997, http://www.emis.de/journals/BAG/vol.38/no.1/1.html.Google Scholar
- A. Császár. A polyhedron without diagonals. Acta Sci. Math. Szeged 13:140--142, 1949.Google Scholar
- G. Di Battista and R. Tamassia. Incremental planarity testing. Proc. 30th Symp. Foundations of Computer Science (FOCS 1989), pp. 436--441, 1989, doi:10.1109/SFCS.1989.63515. Google ScholarDigital Library
- G. Di Battista and R. Tamassia. On-line graph algorithms with SPQR-trees. Proc. 17th Internat. Colloq. Automata, Languages and Programming (ICALP 1990), 443 edition, pp. 598--611. Springer-Verlag, Lecture Notes in Computer Science, 1990, doi:10.1007/BFb0032061. Google ScholarDigital Library
- M. B. Dillencourt and W. D. Smith. Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Math. 161(1-3):63--77, 1996, doi:10.1016/0012-365X(95)00276-3. Google ScholarDigital Library
- M. Donoso and J. O'Rourke. Nonorthogonal polyhedra built from rectangles. Proc. 14th Canadian Conference on Computational Geometry (CCCG'98), 2002, http://arxiv.org/abs/cs/0110059arXiv:cs/0110059.Google Scholar
- V. Dujmović, D. Eppstein, M. Suderman, and D. R. Wood. Drawings of planar graphs with few slopes and segments. Computational Geometry Theory & Applications 38(3):194--212, 2007, doi:10.1016/j.comgeo.2006.09.002, arXiv:math/0606450. Google ScholarDigital Library
- V. Dujmović, M. Suderman, and D. R. Wood. Graph drawings with few slopes. Computational Geometry Theory & Applications 38(3):181--193, 2007, doi:10.1016/j.comgeo.2006.08.002, arXiv:math/0606446. Google ScholarDigital Library
- C. F. Earl and L. J. March. Architectural applications of graph theory. Applications of Graph Theory, pp. 327--355. Academic Press, 1979.Google Scholar
- D. Eppstein. The lattice dimension of a graph. Eur. J. Combinatorics 26(5):585--592, July 2005, doi:10.1016/j.ejc.2004.05.001, arXiv:cs.DS/0402028. Google ScholarDigital Library
- D. Eppstein. Cubic partial cubes from simplicial arrangements. Electronic J. Combinatorics 13(1, R79):1--14, September 2006, arXiv:math.CO/0510263, http://www.combinatorics.org/Volume_13/Abstracts/v13i1r79.html.Google Scholar
- D. Eppstein. Isometric diamond subgraphs. Proc. 16th Int. Symp. Graph Drawing (GD 2008), 5417 edition, pp. 384--389. Springer-Verlag, Lecture Notes in Computer Science, 2008, doi:10.1007/978-3-642-00219-9_37, arXiv:0807.2218. Google ScholarDigital Library
- D. Eppstein. The topology of bendless three-dimensional orthogonal graph drawing. Proc. 16th Int. Symp. Graph Drawing (GD 2008), 5417 edition, pp. 78--89. Springer-Verlag, Lecture Notes in Computer Science, 2008, doi:10.1007/978-3-642-00219-9_9, arXiv:0709.4087. Google ScholarDigital Library
- D. Eppstein and E. Mumford. Orientation-constrained rectangular layouts. Proc. Algorithms and Data Structures Symposium (WADS 2009), 5664 edition, pp. 266--277. Springer-Verlag, Lecture Notes in Computer Science, 2009, arXiv:0904.4312. Google ScholarDigital Library
- D. Eppstein and E. Mumford. Steinitz Theorems for Orthogonal Polyhedra, arXiv:arXiv:0912.0537v1. Electronic preprint, 2010.Google Scholar
- D. Eppstein, E. Mumford, B. Speckmann, and K. A. B. Verbeek. Area-universal rectangular layouts. Proc. 25th ACM Symp. Comp. Geom., pp. 267--276, 2009, doi:10.1145/1542362.1542411, arXiv:0901.3924. Google ScholarDigital Library
- S. Felsner and F. Zickfeld. Schnyder woods and orthogonal surfaces. Discrete and Computational Geometry 40(1):103--126, 2008, doi:10.1007/s00454-007-9027-9.Google ScholarDigital Library
- É. Fusy. Transversal structures on triangulations, with application to straight-line drawing. Proc. 13th Int. Symp. Graph Drawing (GD 2005), 3843 edition, pp. 177--188. Springer-Verlag, Lecture Notes in Computer Science, 2006, doi:10.1007/11618058_17, http://algo.inria.fr/fusy/Articles/FusyGraphDrawing.pdf. Google ScholarDigital Library
- É. Fusy. Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discrete Mathematics 309(7):1870--1894, 2009, doi:10.1016/j.disc.2007.12.093, arXiv:math/0602163.Google ScholarDigital Library
- B. Grünbaum. Convex Polytopes. Graduate Texts in Mathematics. Springer-Verlag, Berlin, Heidelberg, and New York, 2nd edition, 2003.Google Scholar
- C. Gutwenger and P. Mutzel. A linear time implementation of SPQR-trees. Proc. 8th Int. Symp. Graph Drawing (GD 2000), 1984 edition, pp. 77--90. Springer-Verlag, Lecture Notes in Computer Science, 2001. Google ScholarDigital Library
- P. J. Heawood. On the four-colour map theorem. Quarterly J. Pure Appl. Math. 29:270--285, 1898.Google Scholar
- C. D. Hodgson, I. Rivin, and W. D. Smith. A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bull. Amer. Math. Soc. 27:246--251, 1992, doi:10.1090/S0273-0979-1992-00303-8.Google ScholarCross Ref
- S.-H. Hong and H. Nagamochi. Extending Steinitz' Theorem to Non-convex Polyhedra. Tech. Rep. 2008-012, Department of Applied Mathematics & Physics, Kyoto University, 2008, http://www.amp.i.kyoto-u.ac.jp/tecrep/abst/2008/2008-012.html.Google Scholar
- J. Hopcroft and R. Tarjan. Dividing a graph into triconnected components. SIAM J. Comput. 2(3):135--158, 1973, doi:10.1137/0202012.Google ScholarDigital Library
- J. Hopcroft and R. Tarjan. Efficient Planarity Testing. J. ACM 21(4):549--568, 1974, doi:10.1145/321850.321852. Google ScholarDigital Library
- D. A. Huffman. Impossible objects as nonsense sentences. Machine Intelligence 6, pp. 295--323. Edinburgh University Press, 1971.Google Scholar
- G. Kant. Hexagonal grid drawings. Proc. Int. Worksh. Graph-Theoretic Concepts in Computer Science, 657 edition, pp. 263--276. Springer-Verlag, Lecture Notes in Computer Science, 1993, doi:10.1007/3-540-56402-0_53. Google ScholarDigital Library
- G. Kant and X. He. Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoretical Computer Science 172(1-2):175--193, 1997, doi:10.1016/S0304-3975(95)00257-X. Google ScholarDigital Library
- B. Keszegh, J. Pach, D. Pálvölgyi, and G. Tóth. Drawing cubic graphs with at most five slopes. Computational Geometry Theory & Applications 40(2):138--147, 2008, doi:10.1016/j.comgeo.2007.05.003. Google ScholarDigital Library
- L. M. Kirousis and C. H. Papadimitriou. The complexity of recognizing polyhedral scenes. Journal of Computer and System Sciences 37(1):14--38, 1988, doi:10.1016/0022-0000(88)90043-8, http://lca.ceid.upatras.gr/~kirousis/publications/j29.pdf. Google ScholarDigital Library
- K. Kozminski and E. Kinnen. Rectangular duals of planar graphs. Networks 5(2):145--157, 1985, doi:10.1002/net.3230150202.Google ScholarCross Ref
- M. Löffler and E. Mumford. Connected rectilinear graphs on point sets. Proc. 16th Int. Symp. Graph Drawing (GD 2008), 5417 edition, pp. 313--318. Springer-Verlag, Lecture Notes in Computer Science, 2008, doi:10.1007/978-3-642-00219-9_30, http://www.cs.uu.nl/research/techreps/repo/CS-2008/2008-028.pdf.Google Scholar
- S. Mac Lane. A structural characterization of planar combinatorial graphs. Duke Math. J. 3(3):460--472, 1937, doi:10.1215/S0012-7094-37-00336-3.Google ScholarCross Ref
- P. Mukkamala and M. Szegedy. Geometric representation of cubic graphs with four directions. Computational Geometry Theory & Applications 42(9):842--851, 2009, doi:10.1016/j.comgeo.2009.01.005. Google ScholarDigital Library
- J. Pach and D. Pálvölgyi. Bounded degree graphs can have arbitrarily large slope numbers. Electronic Journal of Combinatorics 13(1), 2006, http://www.combinatorics.org/Volume_13/PDF/v13i1n1.pdf.Google ScholarCross Ref
- M. S. Rahman, T. Nishizeki, and M. Naznin. Orthogonal drawings of plane graphs without bends. J. Graph Algorithms & Applications 7(4):335--362, 2003, http://jgaa.info/accepted/2003/Rahman+2003.7.4.pdf.Google ScholarCross Ref
- E. Raisz. The rectangular statistical cartogram. Geographical Review 24(2):292--296, 1934, doi:10.2307/208794.Google ScholarCross Ref
- I. Rinsma. Rectangular and orthogonal floorplans with required rooms areas and tree adjacency. Environment and Planning B: Planning and Design 15:111--118, 1988.Google ScholarCross Ref
- I. Rivin. A characterization of ideal polyhedra in hyperbolic 3-space. Annals of Mathematics 143(1):51--70, 1996, doi:10.2307/2118652.Google ScholarCross Ref
- W. Schnyder. Embedding planar graphs on the grid. Proc. 1st ACM-SIAM Symp. Discrete Algorithms, pp. 138--148, 1990. Google ScholarDigital Library
- E. Steinitz. Polyeder und Raumeinteilungen. Encyclopadie der mathematischen Wissenschaften, Band 3 (Geometries), pp. 1--139, 1922.Google Scholar
- R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3):421--444, 1987, doi:10.1137/0216030. Google ScholarDigital Library
- W. P. Thurston. Conway's tiling groups. Amer. Mathematical Monthly 97(8):757--773, 1990, doi:10.2307/2324578. Google ScholarDigital Library
- D. Waltz. Understanding line drawings of scenes with shadows. The Psychology of Computer Vision, pp. 19--91. McGraw-Hill, 1975.Google Scholar
- D. R. Wood. Lower bounds for the number of bends in three-dimensional orthogonal graph drawings. J. Graph Algorithms & Applications 7(1):33--77, 2003, http://jgaa.info/accepted/2003/Wood2003.7.1.pdf.Google ScholarCross Ref
- D. R. Wood. Optimal three-dimensional orthogonal graph drawing in the general position model. Theoretical Computer Science 299(1-3):151--178, 2003, doi:10.1016/S0304-3975(02)00044-0, http://www.ms.unimelb.edu.au/~woodd/papers/Wood-TCS03.pdf.Google ScholarDigital Library
- G. K. H. Yeap and M. Sarrafzadeh. Sliceable floorplanning by graph dualization. SIAM Journal of Discrete Mathematics 8(2):258--280, 1995, doi:10.1137/S0895480191266700. Google ScholarDigital Library
- G. M. Ziegler. Lectures on Polytopes. Graduate Texts in Mathematics. Springer-Verlag, Berlin, Heidelberg, and New York, 152 edition, 1995.Google Scholar
Index Terms
- Steinitz theorems for orthogonal polyhedra
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