Cubicity and Bandwidth

Abstract

A unit cube in ${\mathbb{R}^k}$ (or a k-cube in short) is defined as the Cartesian product R ₁ × R ₂ × ... × R _k where R _i (for 1 ≤ i ≤ k) is a closed interval of the form [a _i , a _i + 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. From a geometric embedding point of view, a k-cube representation of G = (V, E) yields an embedding ${f: V(G) \rightarrow \mathbb{R}^k}$ such that for any two vertices u and v, ||f(u) − f(v)||_∞ ≤ 1 if and only if ${(u, v) \in E(G)}$ . We first present a randomized algorithm that constructs the cube representation of any graph on n vertices with maximum degree Δ in O(Δ ln n) dimensions. This algorithm is then derandomized to obtain a polynomial time deterministic algorithm that also produces the cube representation of the input graph in the same number of dimensions. The bandwidth ordering of the graph is studied next and it is shown that our algorithm can be improved to produce a cube representation of the input graph G in O(Δ ln b) dimensions, where b is the bandwidth of G, given a bandwidth ordering of G. Note that b ≤ n and b is much smaller than n for many well-known graph classes. Another upper bound of b + 1 on the cubicity of any graph with bandwidth b is also shown. Together, these results imply that for any graph G with maximum degree Δ and bandwidth b, the cubicity is O(min{b, Δ ln b}). The upper bound of b + 1 is used to derive upper bounds for the cubicity of circular-arc graphs, cocomparability graphs and AT-free graphs in terms of the maximum degree Δ.