On the Cubicity of ATFree Graphs and CircularArc Graphs
 L. Sunil Chandran,
 Mathew C. Francis,
 Naveen Sivadasan
 … show all 3 hide
Abstract
A unit cube in k dimensions (kcube) 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 kcubes (cube representation in k dimensions) if each vertex of G can be mapped to a kcube such that two vertices are adjacent in G if and only if their corresponding kcubes have a nonempty intersection. The cubicity of G denoted as cub(G) is the minimum k for which G can be represented as the intersection of kcubes.
An interesting aspect about cubicity is that many problems known to be NPcomplete 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 wellknown graph classes such as ATfree graphs, circulararc graphs and cocomparability graphs which have O(Δ) bandwidth. Thus we have:

cub(G) ≤ 3Δ− 1, if G is an ATfree graph.

cub(G) ≤ 2Δ + 1, if G is a circulararc graph.

cub(G) ≤ 2Δ, if G is a cocomparability graph.
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 Title
 On the Cubicity of ATFree Graphs and CircularArc Graphs
 Book Title
 Graph Theory, Computational Intelligence and Thought
 Book Subtitle
 Essays Dedicated to Martin Charles Golumbic on the Occasion of His 60th Birthday
 Pages
 pp 148157
 Copyright
 2009
 DOI
 10.1007/9783642020292_15
 Print ISBN
 9783642020285
 Online ISBN
 9783642020292
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 5420
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Cubicity
 bandwidth
 intersection graphs
 ATfree graphs
 circulararc graphs
 cocomparability graphs
 Industry Sectors
 eBook Packages
 Editors

 Marina Lipshteyn ^{(16)}
 Vadim E. Levit ^{(17)}
 Ross M. McConnell ^{(18)}
 Editor Affiliations

 16. The Caesarea Rothschild Institute, Mount Carmel, University of Haifa
 17. Department of Computer Science and Mathematics, Ariel University Center of Samaria
 18. Department of Computer Science, Colorado State University
 Authors

 L. Sunil Chandran ^{(19)}
 Mathew C. Francis ^{(19)}
 Naveen Sivadasan ^{(20)}
 Author Affiliations

 19. Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore, 560 012, India
 20. Advanced Technology Centre, TCS, Deccan Park, Madhapur, Hyderabad, 500 081, India
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