On the Cubicity of AT-Free Graphs and Circular-Arc Graphs
A unit cube in k dimensions (k-cube) 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 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.
cub(G) ≤ 3Δ− 1, if G is an AT-free graph.
cub(G) ≤ 2Δ + 1, if G is a circular-arc graph.
cub(G) ≤ 2Δ, if G is a cocomparability graph.
KeywordsCubicity bandwidth intersection graphs AT-free graphs circular-arc graphs cocomparability graphs
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