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Geometric representations of graphs in low dimension

Chandran, Sunil L and Francis, Mathew and Sivadasan, Naveen (2006) Geometric representations of graphs in low dimension. In: 12th Annual International Conference on Computing and Combinatorics (COCOON-2006), , August . 2006, Taiwan, Taipei.

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Official URL: http://arxiv.org/abs/cs/0605013


We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b.

Item Type: Conference Paper
Additional Information: Copyright of this article belongs to Springer-Verlag Berlin.
Keywords: Boxicity;randomized algorithm;derandomization;random graph; intersection graphs
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Ms V Mangala
Date Deposited: 10 Nov 2011 05:54
Last Modified: 10 Nov 2011 05:54
URI: http://eprints.iisc.ac.in/id/eprint/41960

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