Finding a Box Representation for a Graph in O(n2Δ2lnn) Time

Abstract

An axis-parallel box in b-dimensional space is a Cartesian product R₁×R₂×...×R_b where R_i (for 1⩽i⩽b) is a closed interval of the form [a_i, b_i] on the real line. For a graph G, its boxicity is the minimum dimension b, such that G is representable as the intersection graph of (axis-parallel) boxes in b-dimensional space. The concept of boxicity finds application in various areas of research like ecology, operation research etc. Chandran, Francis and Sivadasan gave an O(Δn²ln²n) randomized algorithm to construct a box representation for any graph G on n vertices in [(Δ+2)lnn] dimensions, where ¿ is the maximum degree of the graph. They also came up with a deterministic algorithm that runs in O(n⁴Δ) time. Here, we present an O(n²Δ²lnn) deterministic algorithm that constructs the box representation for any graph in [(Δ+2)lnn] dimensions.