View source for Boxicity

{{Short description|Smallest dimension where a graph can be represented as an intersection graph of boxes}}
[[File:Boxicity.svg|thumb|300px|An intersection graph of rectangles, with boxicity two]]
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 [[box (shape)|box]]es. That is, there must exist a one-to-one correspondence between the [[vertex (graph theory)|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.

==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.

{{harvtxt|Roberts|1969}} showed that the graph with 2''n'' vertices formed by removing a [[perfect matching]] from a [[complete graph]] on 2''n'' 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 2''n'' facets of an ''n''-dimensional [[hypercube]] into a box. Because of these results, this graph has been called the ''Roberts graph'',<ref>E.g., see {{harvtxt|Chandran|Francis|Sivadasan|2010}} and  {{harvtxt|Chandran|Sivadasan|2007}}.</ref> although it is better known as the '''cocktail party graph''' and it can also be understood as the [[Turán graph]] ''T''(2''n'',''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,<ref>{{harvtxt|Scheinerman|1984}}.</ref> and every [[planar graph]] has boxicity at most three.<ref>{{harvtxt|Thomassen|1986}}.</ref>

If a [[bipartite graph]] has boxicity two, it can be represented as an intersection graph of axis-parallel line segments in the plane.<ref>{{harvtxt|Bellantoni|Ben-Arroyo Hartman|Przytycka|Whitesides|1993}}.</ref>

{{harvtxt|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.<ref>{{harvtxt|Chandran|Francis|Sivadasan|2010}} observe that this follows from the fact that these graphs have a polynomial number of [[Maximal clique|maximal cliques]], i.e., the class of graphs with bounded boxicity is said to have [[Graphs with few cliques|few cliques.]] An explicit box representation is not needed to list all maximal cliques efficiently.</ref> For some other graph problems, an efficient solution or approximation can be found if a low-dimensional box representation is known.<ref>See, e.g., {{harvtxt|Agarwal|van Kreveld|Suri|1998}} and {{harvtxt|Berman|DasGupta|Muthukrishnan|Ramaswami|2001}} for approximations to the [[maximum independent set]] for intersection graphs of rectangles, and {{harvtxt|Chlebík|Chlebíková|2005}} for results on hardness of approximation of these problems in higher dimensions.</ref> 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.<ref>{{harvtxt|Cozzens|1981}} shows that computing the boxicity is NP-complete; {{harvtxt|Yannakakis|1982}} shows that even checking whether the boxicity is at most 3 is NP-hard; finally {{harvtxt|Kratochvil|1994}} showed that recognising boxicity 2 is NP-hard.</ref>
{{harvtxt|Chandran|Francis|Sivadasan|2010}} describe algorithms for finding representations of arbitrary graphs as intersection graphs of boxes, with a dimension that is within a [[logarithm]]ic factor of the [[degree (graph theory)|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 [[parameterized complexity|fixed-parameter tractable]] when parameterized by the [[vertex cover]] number of the input graph.<ref>{{harvtxt|Adiga|Chitnis|Saurabh|2010}}.</ref>

==Bounds==
If a graph ''G'' has [[treewidth]] {{math|tw}}, then:
<math>\operatorname{box}(G) \le \mathrm{tw} + 2</math>.<ref>{{harvtxt|Chandran|Sivadasan|2007|loc=Theorem 14}}</ref>

If a graph ''G'' has ''m'' edges, then:
<math>\operatorname{box}(G) = O(\sqrt{m \cdot \log(m)})</math>.<ref>{{harvtxt|Chandran|Francis|Sivadasan|2010}}</ref><ref>{{harvtxt|Esperet|2016}}</ref>

If a graph ''G'' is ''k''-[[Degeneracy (graph theory)|degenerate]] (with <math>k\ge 2</math>) and has ''n'' vertices, then ''G'' has boxicity <math>\operatorname{box}(G) \le (k+2) \lceil 2e \log n \rceil</math>.<ref>{{harvtxt|Adiga|Chandran|Mathew|2014}}</ref>

If a graph ''G'' has no complete graph on ''t'' vertices as a [[Graph minor|minor]], then <math>\operatorname{box}(G) = O(t^2 \log t)</math><ref>{{harvtxt|Esperet|Wiechert|2018}}</ref> while there are graphs with no complete graph on ''t'' vertices as a [[Graph minor|minor]], and with boxicity <math>\Omega(t \sqrt{\log t})</math>.<ref>{{harvtxt|Esperet|2016}}</ref> In particular, any graph ''G'' has boxicity <math>\operatorname{box}(G) = O(\mu(G)^2 \log \mu(G))</math>, where <math>\mu(G)</math> denotes the [[Colin de Verdière graph invariant|Colin de Verdière invariant]] of ''G''.

==Related graph invariants==
* [[Cubicity]] is defined in the same way as boxicity but with axis-parallel [[hypercube]]s instead of hyperrectangles.<!--For a graph G, its cubicity cub(G) is the minimum dimension k such that G is representable as the intersection graph of (axis-parallel) cubes in k-dimensional space.--> Boxicity is a generalization of cubicity.
* [[Sphericity (graph theory)|Sphericity]] is defined in the same way as boxicity but with unit-diameter spheres.<!-- The sphericity of a graph G is the smallest n such that the vertices in G can be mapped into unit-diameter spheres in n-space in such a way that two vertices are adjacent in G if and only if their assigned spheres intersect.-->



==Notes==
{{reflist}}

==References==
{{refbegin|30em}}
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{{refend}}

[[Category:Geometric graph theory]]
[[Category:Graph invariants]]
[[Category:NP-complete problems]]