On the Complexity of Simultaneous Geometric Embedding for Edge-Disjoint Graphs (2312.09025v1)
Abstract: Simultaneous Geometric Embedding (SGE) asks whether, for a given collection of graphs on the same vertex set V, there is an embedding of V in the plane that admits a crossing-free drawing with straightline edges for each of the given graphs. It is known that SGE is $\exists\mathbb{R}$-complete, that is, the problem is polynomially equivalent to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution. We prove that SGE remains $\exists\mathbb{R}$-complete for edge-disjoint input graphs, that is, for collections of graphs without so-called public edges. As an intermediate result, we prove that it is $\exists\mathbb{R}$-complete to decide whether a directional walk without repeating edges is realizable. Here, a directional walk consists of a sequence of not-necessarily distinct vertices (a walk) and a function prescribing for each inner position whether the walk shall turn left or shall turn right. A directional walk is realizable, if there is an embedding of its vertices in the plane such that the embedded walk turns according to the given directions. Previously it was known that realization is $\exists\mathbb{R}$-complete to decide for directional walks repeating each edge at most 336 times. This answers two questions posed by Schaefer ["On the Complexity of Some Geometric Problems With Fixed Parameters", JGAA 2021].
- The art gallery problem is ∃ℝℝ\exists\mathbb{R}∃ blackboard_R-complete. J. ACM, 69(1):4:1–4:70, 2022. doi:10.1145/3486220.
- Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph. J. Discrete Algorithms, 14:150–172, 2012. doi:10.1016/j.jda.2011.12.015.
- On a tree and a path with no geometric simultaneous embedding. J. Graph Algorithms Appl., 16(1):37–83, 2012. doi:10.7155/jgaa.00250.
- Advancements on SEFE and partitioned book embedding problems. Theor. Comput. Sci., 575:71–89, 2015. doi:10.1016/j.tcs.2014.11.016.
- Simultaneous embedding of planar graphs. In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization, pages 349–381. Chapman and Hall/CRC, 2013.
- On the classification of non-realizable oriented matroids (part I: generation). Preprint, 1990.
- On simultaneous planar graph embeddings. Comput. Geom., 36(2):117–130, 2007. doi:10.1016/j.comgeo.2006.05.006.
- John F. Canny. Some algebraic and geometric computations in PSPACE. In Janos Simon, editor, 20th Annual ACM Symposium on Theory of Computing, STOC 1988, pages 460–467. ACM, 1988. doi:10.1145/62212.62257.
- Jean Cardinal. Computational Geometry Column 62. SIGACT News, 46(4):69–78, 2015. doi:10.1145/2852040.2852053.
- The complexity of simultaneous geometric graph embedding. J. Graph Algorithms Appl., 19(1):259–272, 2015. doi:10.7155/jgaa.00356.
- Thickness and colorability of geometric graphs. Comput. Geom., 56:1–18, 2016. doi:10.1016/j.comgeo.2016.03.003.
- Parameterized complexity of simultaneous planarity. In Michael A. Bekos and Markus Chimani, editors, Graph Drawing and Network Visualization, 31st International Symposium, GD 2023, volume to appear of LNCS. Springer, 2023.
- Fabrizio Frati. Embedding graphs simultaneously with fixed edges. In Michael Kaufmann and Dorothea Wagner, editors, Graph Drawing, 14th International Symposium, GD 2006, volume 4372 of LNCS, pages 108–113. Springer, 2007. doi:10.1007/978-3-540-70904-6_12.
- Geometric thickness of multigraphs is ∃ℝℝ\exists\mathbb{R}∃ blackboard_R-complete. Preprint, 2023. arXiv:2312.05010.
- Simultaneous graph embeddings with fixed edges. In Fedor V. Fomin, editor, Graph-Theoretic Concepts in Computer Science, WG 2006, volume 4271 of LNCS, pages 325–335. Springer, 2006. doi:10.1007/11917496_29.
- Testing simultaneous planarity when the common graph is 2-connected. J. Graph Algorithms Appl., 17(3):147–171, 2013. doi:10.7155/jgaa.00289.
- Donald E. Knuth. Axioms and Hulls, volume 606 of LNCS. Springer, 1992. doi:10.1007/3-540-55611-7.
- Jan Kyncl. Simple realizability of complete abstract topological graphs in P. Discret. Comput. Geom., 45(3):383–399, 2011. doi:10.1007/s00454-010-9320-x.
- Jirí Matousek. Intersection graphs of segments and ∃ℝℝ\exists\mathbb{R}∃ blackboard_R. Preprint, 2014. arXiv:1406.2636.
- Nikolai E. Mnëv. The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In Topology and Geometry — Rohlin Seminar, volume 1346 of Lecture Notes in Mathematics, pages 527–543. Springer, 1988. doi:10.1007/BFb0082792.
- Embedding planar graphs at fixed vertex locations. Graphs Comb., 17(4):717–728, 2001. doi:10.1007/PL00007258.
- Oriented matroids. In Jacob E. Goodman and Joseph O’Rourke, editors, Handbook of Discrete and Computational Geometry, Second Edition, pages 129–151. Chapman and Hall/CRC, 2004. doi:10.1201/9781420035315.CH6.
- Ignaz Rutter. Simultaneous embedding. In Seok-Hee Hong and Takeshi Tokuyama, editors, Beyond Planar Graphs, Communications of NII Shonan Meetings, pages 237–265. Springer, 2020. doi:10.1007/978-981-15-6533-5_13.
- Marcus Schaefer. Complexity of some geometric and topological problems. In Graph Drawing, 17th International Symposium, GD 2009, volume 5849 of LNCS, pages 334–344. Springer, 2009. doi:10.1007/978-3-642-11805-0_32.
- Marcus Schaefer. On the complexity of some geometric problems with fixed parameters. J. Graph Algorithms Appl., 25(1):195–218, 2021. doi:10.7155/jgaa.00557.
- Raphael Steiner. A logarithmic bound for simultaneous embeddings of planar graphs. In Michael A. Bekos and Markus Chimani, editors, Graph Drawing and Network Visualization, 31st International Symposium, GD 2023, volume to appear of LNCS. Springer, 2023.
- Falk Tschirschnitz. Testing extendability for partial chirotopes is NP-complete. In 13th Canadian Conference on Computational Geometry, CCCG 2001, pages 165–168, 2001.