Simultaneous embedding

In graph drawing and information visualization, simultaneous embedding is a technique for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed; it is only crossings between two edges of the same graph that are disallowed.[1]

If edges are allowed to be drawn as polylines or curves, then any planar graph may be drawn without crossings with its vertices in arbitrary positions in the plane. Using the same vertex placement for two graphs provides a simultaneous embedding of the two graphs. Research in this type of simultaneous embedding has concentrated on finding drawings with few bends, or with few crossings between edges from the two graphs.[1] Restricted models of simultaneous embedding have also been studied. In simultaneous geometric embedding, each graph must be drawn planarly with edges representing its edges rather than more complex curves. In order to guarantee that such an embedding exists, one must restrict the two given graphs to subclasses of the planar graphs. In another restricted model of simultaneous embedding, simultaneous embedding with fixed edges, curves or bends are allowed in the edges but any edge that is present in both of the given graphs must be represented by the same curve in both drawings. When a simultaneous geometric embedding exists, it automatically is also a simultaneous embedding with fixed edges.[1]

Simultaneous embedding is closely related to thickness, the minimum number of planar subgraphs that can cover all of the edges of a given graph, and geometric thickness, the minimum number of edge colors needed in a straight-line drawing of a given graph with no crossing between edges of the same color. In particular, the thickness of a given graph is two if the graph's edges can be partitioned into two subgraphs that have a simultaneous embedding, and the geometric thickness is two if the edges can be partitioned into two subgraphs that have a simultaneous geometric embedding.

For simultaneous embedding problems on more than two graphs, it is standard to assume that all pairs of input graphs have the same intersection as each other; that is, the edge and vertex sets of the graphs form a sunflower. This constraint is known as sunflower intersection.[1]

Simultaneous geometric embedding

Many results on simultaneous geometric embedding are based on the idea that the Cartesian coordinates of the two given graphs' vertices can be derived from properties of the two graphs. One of the most basic results of this type is the fact that any two path graphs on the same vertex set always have a simultaneous embedding. To find such an embedding, one can use the position of a vertex in the first path as its x-coordinate, and the position of the same vertex in the second path as its y-coordinate. In this way, the first path will be drawn as an x-monotone polyline, a type of curve that is automatically non-self-crossing, and the second path will similarly be drawn as a y-monotone polyline.[2]

This type of drawing places the vertices in an integer lattice of dimensions linear in the graph sizes. Similarly defined layouts also work, with larger but still linear grid sizes, when both graphs are caterpillars or when both are cycle graphs. A simultaneous embedding in a grid of linear dimensions is also possible for any number of graphs that are all stars. Other pairs of graph types that always admit a simultaneous embedding, but that might need larger grid sizes, include a wheel graph and a cycle graph, a tree and a matching, or a pair of graphs both of which have maximum degree two. However, there exist pairs of a planar graphs and a matching, or of a tree and a path, that have no simultaneous geometric embedding.[3][4][5]

Testing whether two graphs admit a simultaneous geometric embedding is NP-hard.[1][6] More precisely, it is complete for the existential theory of the reals. The proof of this result also implies that for some pairs of graphs that have simultaneous geometric embeddings, the smallest grid on which they can be drawn has doubly exponential size.[7][8]

Simultaneous embedding with fixed edges

For simultaneous embedding with fixed edges, the classification of different types of input as always having an embedding or as sometimes not being possible depends not only on the two types of graphs to be drawn, but also on the structure of their intersection. For instance, it is always possible to find such an embedding when both of the two given graphs are outerplanar graphs and their intersection is a linear forest, with at most one bend per edge and with vertex coordinates and bend points all belonging to a grid of polynomial area. However, there exist other pairs of outerplanar graphs with more complex intersections that have no such embedding. It is also possible to find a simultaneous embedding with fixed edges for any pair of a planar graph and a tree.[9][10][11]

Unsolved problem in mathematics:
Can a simultaneous embedding with fixed edges for two given graphs be found in polynomial time?
(more unsolved problems in mathematics)

It is an open question whether the existence of a simultaneous embedding with fixed edges for two given graphs can be tested in polynomial time. However, for three or more graphs, the problem is NP-complete. Simultaneous embeddings with fixed edges can be found (when they exist) in polynomial time for pairs of outerplanar graphs, and for pairs of graphs whose intersection is biconnected.[1][12][13][14]

Unrestricted simultaneous embedding

Any two planar graphs have a simultaneous embedding. This may be done in a grid of polynomial area, with at most two bends per edge. Any two subhamiltonian graphs have a simultaneous embedding with at most one bend per edge.[1][10][15]

References

  1. 1 2 3 4 5 6 7 Bläsius, Thomas; Kobourov, Stephen G.; Rutter, Ignaz (2013), "Simultaneous embedding of planar graphs", in Tamassia, Roberto, Handbook of Graph Drawing and Visualization, CRC Press, pp. 349–383, ISBN 9781420010268
  2. ↑ Bläsius, Kobourov & Rutter (2013), Theorem 11.1.
  3. ↑ Bläsius, Kobourov & Rutter (2013), Figure 11.2.
  4. ↑ Brass, Peter; Cenek, Eowyn; Duncan, Christian A.; Efrat, Alon; Erten, Cesim; Ismailescu, Dan P.; Kobourov, Stephen G.; Lubiw, Anna; Mitchell, Joseph S. B. (2007), "On simultaneous planar graph embeddings", Computational Geometry Theory & Applications, 36 (2): 117–130, doi:10.1016/j.comgeo.2006.05.006, MR 2278011.
  5. ↑ Cabello, Sergio; van Kreveld, Marc; Liotta, Giuseppe; Meijer, Henk; Speckmann, Bettina; Verbeek, Kevin (2011), "Geometric simultaneous embeddings of a graph and a matching", Journal of Graph Algorithms and Applications, 15 (1): 79–96, doi:10.7155/jgaa.00218, MR 2776002.
  6. ↑ Estrella-Balderrama, Alejandro; Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2008), "Simultaneous geometric graph embeddings", Graph Drawing: 15th International Symposium, GD 2007, Sydney, Australia, September 24–26, 2007, Revised Papers, Lecture Notes in Computer Science, 4875, Berlin: Springer, pp. 280–290, doi:10.1007/978-3-540-77537-9_28, MR 2427826.
  7. ↑ Cardinal, Jean; Kusters, Vincent (2015), "The complexity of simultaneous geometric graph embedding", Journal of Graph Algorithms and Applications, 19 (1): 259–272, MR 3344782.
  8. ↑ Duncan, Christian; Eppstein, David; Kobourov, Stephen G. (2004), "The geometric thickness of low degree graphs", Proc. 20th ACM Symposium on Computational Geometry, ACM, pp. 340–346, arXiv:cs.CG/0312056Freely accessible, doi:10.1145/997817.997868.
  9. ↑ Bläsius, Kobourov & Rutter (2013), Figure 11.5.
  10. 1 2 Di Giacomo, Emilio; Liotta, Giuseppe (2007), "Simultaneous embedding of outerplanar graphs, paths, and cycles", International Journal of Computational Geometry & Applications, 17 (2): 139–160, doi:10.1142/S0218195907002276, MR 2309902.
  11. ↑ Frati, Fabrizio (2007), "Embedding graphs simultaneously with fixed edges", Graph Drawing: 14th International Symposium, GD 2006, Karlsruhe, Germany, September 18–20, 2006, Revised Papers, Lecture Notes in Computer Science, 4372, Berlin: Springer, pp. 108–113, doi:10.1007/978-3-540-70904-6_12, MR 2393910.
  12. ↑ Fowler, J. Joseph; Jünger, Michael; Kobourov, Stephen G.; Schulz, Michael (2011), "Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges", Computational Geometry Theory & Applications, 44 (8): 385–398, doi:10.1016/j.comgeo.2011.02.002, MR 2805957.
  13. ↑ Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2006), "Simultaneous graph embeddings with fixed edges", Graph-Theoretic Concepts in Computer Science: 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers, Lecture Notes in Computer Science, 4271, Berlin: Springer, pp. 325–335, doi:10.1007/11917496_29, MR 2290741.
  14. ↑ Haeupler, Bernhard; Jampani, Krishnam Raju; Lubiw, Anna (2013), "Testing simultaneous planarity when the common graph is 2-connected", Journal of Graph Algorithms and Applications, 17 (3): 147–171, doi:10.7155/jgaa.00289, MR 3043207.
  15. ↑ Di Giacomo, Emilio; Liotta, Giuseppe (2005), "A note on simultaneous embedding of planar graphs", 21st European Workshop on Computational Geometry (PDF), Eindhoven University of Technology.
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