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Boxicity
Minimum dimension in which a given graph can be represented as intersection graph of axis-parallel boxes

In graph theory, boxicity is a graph invariant, introduced by Fred S. Roberts in 1969.

The boxicity of a graph is the minimum dimension in which a given graph can be represented as an intersection graph of axis-parallel boxes. That is, there must exist a one-to-one correspondence between the vertices of the graph and a set of boxes, such that two boxes intersect if and only if there is an edge connecting the corresponding vertices.

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Examples

The figure shows a graph with six vertices, and a representation of this graph as an intersection graph of rectangles (two-dimensional boxes). This graph cannot be represented as an intersection graph of boxes in any lower dimension, so its boxicity is two.

Roberts (1969) showed that the graph with 2n vertices formed by removing a perfect matching from a complete graph on 2n vertices has boxicity exactly n: each pair of disconnected vertices must be represented by boxes that are separated in a different dimension than each other pair. A box representation of this graph with dimension exactly n can be found by thickening each of the 2n facets of an n-dimensional hypercube into a box. Because of these results, this graph has been called the Roberts graph,1 although it is better known as the cocktail party graph and it can also be understood as the Turán graph T(2n,n).

Relation to other graph classes

A graph has boxicity at most one if and only if it is an interval graph; the boxicity of an arbitrary graph G is the minimum number of interval graphs on the same set of vertices such that the intersection of the edges sets of the interval graphs is G. Every outerplanar graph has boxicity at most two,2 and every planar graph has boxicity at most three.3

If a bipartite graph has boxicity two, it can be represented as an intersection graph of axis-parallel line segments in the plane.4

Adiga, Bhowmick & Chandran (2011) proved that the boxicity of a bipartite graph G is within of a factor 2 of the order dimension of the height-two partially ordered set associated to G as follows: the set of minimal elements corresponds to one partite set of G, the set of maximal elements corresponds to the second partite set of G, and two elements are comparable if the corresponding vertices are adjacent in G. Equivalently, the order dimension of a height-two partially ordered set P is within a factor 2 of the boxicity of the comparability graph of P (which is bipartite, since P has height two).

Algorithmic results

Many graph problems can be solved or approximated more efficiently for graphs with bounded boxicity than they can for other graphs; for instance, the maximum clique problem can be solved in polynomial time for graphs with bounded boxicity.5 For some other graph problems, an efficient solution or approximation can be found if a low-dimensional box representation is known.6 However, finding such a representation may be difficult: it is NP-complete to test whether the boxicity of a given graph is at most some given value K, even for K = 2.7 Chandran, Francis & Sivadasan (2010) describe algorithms for finding representations of arbitrary graphs as intersection graphs of boxes, with a dimension that is within a logarithmic factor of the maximum degree of the graph; this result provides an upper bound on the graph's boxicity.

Despite being hard for its natural parameter, boxicity is fixed-parameter tractable when parameterized by the vertex cover number of the input graph.8

Bounds

If a graph G has treewidth tw, then: box ⁡ ( G ) ≤ t w + 2 {\displaystyle \operatorname {box} (G)\leq \mathrm {tw} +2} .9

If a graph G has m edges, then: box ⁡ ( G ) = O ( m ⋅ log ⁡ ( m ) ) {\displaystyle \operatorname {box} (G)=O({\sqrt {m\cdot \log(m)}})} .1011

If a graph G is k-degenerate (with k ≥ 2 {\displaystyle k\geq 2} ) and has n vertices, then G has boxicity box ⁡ ( G ) ≤ ( k + 2 ) ⌈ 2 e log ⁡ n ⌉ {\displaystyle \operatorname {box} (G)\leq (k+2)\lceil 2e\log n\rceil } .12

If a graph G has no complete graph on t vertices as a minor, then box ⁡ ( G ) = O ( t 2 log ⁡ t ) {\displaystyle \operatorname {box} (G)=O(t^{2}\log t)} 13 while there are graphs with no complete graph on t vertices as a minor, and with boxicity Ω ( t log ⁡ t ) {\displaystyle \Omega (t{\sqrt {\log t}})} .14 In particular, any graph G has boxicity box ⁡ ( G ) = O ( μ ( G ) 2 log ⁡ μ ( G ) ) {\displaystyle \operatorname {box} (G)=O(\mu (G)^{2}\log \mu (G))} , where μ ( G ) {\displaystyle \mu (G)} denotes the Colin de Verdière invariant of G.

  • Cubicity is defined in the same way as boxicity but with axis-parallel hypercubes instead of hyperrectangles. Boxicity is a generalization of cubicity.
  • Sphericity is defined in the same way as boxicity but with unit-diameter spheres.

Notes

References

  1. E.g., see Chandran, Francis & Sivadasan (2010) and Chandran & Sivadasan (2007). - Chandran, L. Sunil; Francis, Mathew C.; Sivadasan, Naveen (2010), "Geometric representation of graphs in low dimension using axis parallel boxes", Algorithmica, 56 (2): 129–140, arXiv:cs.DM/0605013, doi:10.1007/s00453-008-9163-5, MR 2576537, S2CID 17838951 https://arxiv.org/abs/cs.DM/0605013 ↩

  2. Scheinerman (1984). - Scheinerman, E. R. (1984), Intersection Classes and Multiple Intersection Parameters, Ph. D thesis, Princeton University https://archive.org/details/intersection-classes-and-multi ↩

  3. Thomassen (1986). - Thomassen, Carsten (1986), "Interval representations of planar graphs", Journal of Combinatorial Theory, Series B, 40: 9–20, doi:10.1016/0095-8956(86)90061-4 https://doi.org/10.1016%2F0095-8956%2886%2990061-4 ↩

  4. Bellantoni et al. (1993). - Bellantoni, Stephen J.; Ben-Arroyo Hartman, Irith; Przytycka, Teresa; Whitesides, Sue (1993), "Grid intersection graphs and boxicity", Discrete Mathematics, 114 (1–3): 41–49, doi:10.1016/0012-365X(93)90354-V https://doi.org/10.1016%2F0012-365X%2893%2990354-V ↩

  5. Chandran, Francis & Sivadasan (2010) observe that this follows from the fact that these graphs have a polynomial number of maximal cliques, i.e., the class of graphs with bounded boxicity is said to have few cliques. An explicit box representation is not needed to list all maximal cliques efficiently. - Chandran, L. Sunil; Francis, Mathew C.; Sivadasan, Naveen (2010), "Geometric representation of graphs in low dimension using axis parallel boxes", Algorithmica, 56 (2): 129–140, arXiv:cs.DM/0605013, doi:10.1007/s00453-008-9163-5, MR 2576537, S2CID 17838951 https://arxiv.org/abs/cs.DM/0605013 ↩

  6. See, e.g., Agarwal, van Kreveld & Suri (1998) and Berman et al. (2001) for approximations to the maximum independent set for intersection graphs of rectangles, and Chlebík & Chlebíková (2005) for results on hardness of approximation of these problems in higher dimensions. - Agarwal, Pankaj K.; van Kreveld, Marc; Suri, Subhash (1998), "Label placement by maximum independent set in rectangles", Computational Geometry Theory and Applications, 11 (3–4): 209–218, doi:10.1016/S0925-7721(98)00028-5, hdl:1874/18908, S2CID 2381363 https://doi.org/10.1016%2FS0925-7721%2898%2900028-5 ↩

  7. Cozzens (1981) shows that computing the boxicity is NP-complete; Yannakakis (1982) shows that even checking whether the boxicity is at most 3 is NP-hard; finally Kratochvil (1994) showed that recognising boxicity 2 is NP-hard. - Cozzens, M. B. (1981), Higher and Multidimensional Analogues of Interval Graphs, Ph.D. thesis, Rutgers University ↩

  8. Adiga, Chitnis & Saurabh (2010). - Adiga, Abhijin; Chitnis, Rajesh; Saurabh, Saket (2010), "Parameterized algorithms for boxicity", Algorithms and Computation: 21st International Symposium, ISAAC 2010, Jeju Island, Korea, December 15-17, 2010, Proceedings, Part I (PDF), Lecture Notes in Computer Science, vol. 6506, pp. 366–377, doi:10.1007/978-3-642-17517-6_33, ISBN 978-3-642-17516-9, archived from the original (PDF) on 2017-08-30, retrieved 2018-01-22 https://web.archive.org/web/20170830050212/http://www.wisdom.weizmann.ac.il/~chitnis/fptBoxicity.pdf ↩

  9. Chandran & Sivadasan (2007, Theorem 14) - Chandran, L. Sunil; Sivadasan, Naveen (2007), "Boxicity and treewidth", Journal of Combinatorial Theory, Series B, 97 (5): 733–744, arXiv:math.CO/0505544, doi:10.1016/j.jctb.2006.12.004, S2CID 9854207 https://arxiv.org/abs/math.CO/0505544 ↩

  10. Chandran, Francis & Sivadasan (2010) - Chandran, L. Sunil; Francis, Mathew C.; Sivadasan, Naveen (2010), "Geometric representation of graphs in low dimension using axis parallel boxes", Algorithmica, 56 (2): 129–140, arXiv:cs.DM/0605013, doi:10.1007/s00453-008-9163-5, MR 2576537, S2CID 17838951 https://arxiv.org/abs/cs.DM/0605013 ↩

  11. Esperet (2016) - Esperet, Louis (2016), "Boxicity and topological invariants", European Journal of Combinatorics, 51: 495–499, arXiv:1503.05765, doi:10.1016/j.ejc.2015.07.020, S2CID 5548385 https://arxiv.org/abs/1503.05765 ↩

  12. Adiga, Chandran & Mathew (2014) - Adiga, Abhijin; Chandran, L. Sunil; Mathew, Rogers (2014), "Cubicity, Degeneracy, and Crossing Number", European Journal of Combinatorics, 35: 2–12, arXiv:1105.5225, doi:10.1016/j.ejc.2013.06.021, S2CID 2069078 https://arxiv.org/abs/1105.5225 ↩

  13. Esperet & Wiechert (2018) - Esperet, Louis; Wiechert, Veit (2018), "Boxicity, poset dimension, and excluded minors", Electronic Journal of Combinatorics, 25 (4): #P4.51, arXiv:1804.00850, doi:10.37236/7787, S2CID 119148637 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i4p51 ↩

  14. Esperet (2016) - Esperet, Louis (2016), "Boxicity and topological invariants", European Journal of Combinatorics, 51: 495–499, arXiv:1503.05765, doi:10.1016/j.ejc.2015.07.020, S2CID 5548385 https://arxiv.org/abs/1503.05765 ↩