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

• Authors: L. Sunil Chandran;Mathew C. Francis;Naveen Sivadasan
• Venue: Graph Theory, Computational Intelligence and Thought
• Year: 2009

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