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# Finding a Box Representation for a Graph in $O(n^2\Delta^2\ln n)$ time

Chandran, Sunil L and Francis, Mathew C and Mathew, Rogers (2008) Finding a Box Representation for a Graph in $O(n^2\Delta^2\ln n)$ time. In: ICIT '08 Proceedings of the 2008 International Conference on Information Technology, 17-20 Dec. 2008 , Washington, DC.

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## Abstract

An axis-parallel box in $b$-dimensional space is a Cartesian product $R_1 \times R_2 \times \cdots \times R_b$ where $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its boxicity is the minimum dimension $b$, such that $G$ is representable as the intersection graph of (axis-parallel) boxes in $b$-dimensional space. The concept of boxicity finds application in various areas of research like ecology, operation research etc. Chandran, Francis and Sivadasan gave an $O(\Delta n^2 \ln^2 n)$ randomized algorithm to construct a box representation for any graph $G$ on $n$ vertices in $\lceil (\Delta + 2)\ln n \rceil$ dimensions, where $\Delta$ is the maximum degree of the graph. They also came up with a deterministic algorithm that runs in $O(n^4 \Delta )$ time. Here, we present an $O(n^2 \Delta^2 \ln n)$ deterministic algorithm that constructs the box representation for any graph in $\lceil (\Delta + 2)\ln n \rceil$ dimensions.

Item Type: Conference Paper Copyright 2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 23 Sep 2011 09:25 23 Sep 2011 09:25 http://eprints.iisc.ernet.in/id/eprint/40693

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