On the Cubicity of AT-Free Graphs and Circular-Arc Graphs
- L. Sunil ChandranAffiliated withDept. of Computer Science and Automation, Indian Institute of Science
- , Mathew C. FrancisAffiliated withDept. of Computer Science and Automation, Indian Institute of Science
- , Naveen SivadasanAffiliated withAdvanced Technology Centre, TCS
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.
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
- On the Cubicity of AT-Free Graphs and Circular-Arc Graphs
- Book Title
- Graph Theory, Computational Intelligence and Thought
- Book Subtitle
- Essays Dedicated to Martin Charles Golumbic on the Occasion of His 60th Birthday
- pp 148-157
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
- Additional Links
- intersection graphs
- AT-free graphs
- circular-arc graphs
- cocomparability graphs
- Industry Sectors
- 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
- 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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