Boxicity and poset dimension

Abstract

Let G be a simple, undirected, finite graph with vertex set V(G) and edge set E(G). A k-dimensional box is a Cartesian product of closed intervals [a 1,b 1]×[a 2,b 2] ×⋯×[a k ,b k ]. The boxicity of G, box(G) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes, i.e. each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let be a poset where S is the ground set and P is a reflexive, anti-symmetric and transitive binary relation on S. The dimension of , is the minimum integer t such that P can be expressed as the intersection of t total orders. Let be the underlying comparability graph of . It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset , , where is the chromatic number of and . The second result of the paper relates the boxicity of a graph G with a natural partial order associated with its extended double cover, denoted as G c . Let be the natural height-2 poset associated with G c by making A the set of minimal elements and B the set of maximal elements. We show that . These results have some immediate and significant consequences. The upper bound allows us to derive hitherto unknown upper bounds for poset dimension. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree Δ is O(Δlog2Δ) which is an improvement over the best known upper bound of Δ2+2. (2) There exist graphs with boxicity Ω(ΔlogΔ). This disproves a conjecture that the boxicity of a graph is O(Δ). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n 0.5-ε ) for any ε>0, unless NP=ZPP. © 2010 Springer-Verlag Berlin Heidelberg.