Representing graphs as the intersection of axis-parallel cubes

A unit cube in k dimensional space (or k-cube in short) is defined as the Cartesian product R1 × R2 × · · · × Rk where Ri(for 1 ≤ i ≤ k) is a closed interval of the form [ai, ai + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G is the minimum k such that G has a k-cube representation. Many NPcomplete graph problems have polynomial time deterministic algorithms or have good approximation ratios in graphs of low cubicity. In most of these algorithms, computing a low dimensional cube representation of the given graph is usually the first step. From a geometric embedding point of view, a k-cube representation of G = (V,E) yields an embedding f : V → R such that for any two vertices u and v, ||f(u) − f(v)||∞ ≤ 1 if and only if (u, v) ∈ E. The best known upper bound is a k-cube representation of any given graph G on n nodes in k ≤ 2n/3 dimension. Our main result is an efficient algorithm to compute the k-cube representation of G with maximum degree ∆ in k = O(min{bw,∆ ln(bw)}) dimensions, given a bandwidth ordering of G with width bw. Note that bw ≤ n and bw is much smaller than n for many well-known graph classes. Though computing the bandwidth ordering is NP-hard in general, using the known approximation algorithms for it, our result imply efficient algorithms to compute: (a) k-cube representation of G with k = O(∆(ln(bw) + ln lnn)) for any graph G; (b) k = O(1) for bounded bandwidth graphs and (c)k = O(∆) for many well-known graph classes like circular arc graphs and AT-free graphs. We show graphs where our general upper bound on k is tight upto a factor of O(ln lnn). Along with other structural results, we also show that for almost all graphs, the cubicity is upper bound by O(dav lnn), where dav is the average degree.