On the Cubicity of AT-Free Graphs and Circular-Arc Graphs

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Abstract

A unit cube in k dimensions (k-cube) 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 graph G on n nodes is said to be representable as the intersection of k-cubes (cube representation in k dimensions) if each vertex of G can be mapped to a k-cube such that two vertices are adjacent in G if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of k-cubes.

An interesting aspect about cubicity is that many problems known to be NP-complete for general graphs 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.

We give an O(bw·n) algorithm to compute the cube representation of a general graph G in bw + 1 dimensions given a bandwidth ordering of the vertices of G, where bw is the bandwidth of G. As a consequence, we get O(Δ) upper bounds on the cubicity of many well-known graph classes such as AT-free graphs, circular-arc graphs and cocomparability graphs which have O(Δ) bandwidth. Thus we have:

  1. cub(G) ≤ 3Δ− 1, if G is an AT-free graph.

  2. cub(G) ≤ 2Δ + 1, if G is a circular-arc graph.

  3. cub(G) ≤ 2Δ, if G is a cocomparability graph.

Also for these graph classes, there are constant factor approximation algorithms for bandwidth computation that generate orderings of vertices with O(Δ) width. We can thus generate the cube representation of such graphs in O(Δ) dimensions in polynomial time.