Cubicity, degeneracy, and crossing number

Adiga, Abhijin and Chandran, Sunil L and Mathew, Rogers (2011) Cubicity, degeneracy, and crossing number. In: Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2011, 01.12.2011.

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Abstract

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as $\cubi(G)$, is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/ 0607092, 2006.] that, for a graph $G$ with maximum degree $\Delta$, $\cubi(G)\leq \lceil 4(\Delta +1)\log n\rceil$. In this paper, we show that, for a $k$-degenerate graph $G$, $\cubi(G) \leq (k+2) \lceil 2e \log n \rceil$. Since $k$ is at most $\Delta$ and can be much lower, this clearly is a stronger result. This bound is tight. We also give an efficient deterministic algorithm that runs in $O(n^2k)$ time to output a $8k(\lceil 2.42 \log n\rceil + 1)$ dimensional cube representation for $G$. An important consequence of the above result is that if the crossing number of a graph $G$ is $t$, then $\boxi(G)$ is $O(t^{1/4}{\lceil\log t\rceil}^{3/4})$ . This bound is tight up to a factor of $O((\log t)^{1/4})$. We also show that, if $G$ has $n$ vertices, then $\cubi(G)$ is $O(\log n + t^{1/4}\log t)$. Using our bound for the cubicity of $k$-degenerate graphs we show that cubicity of almost all graphs in $\mathcal{G}(n,m)$ model is $O(d_{av}\log n)$, where $d_{av}$ denotes the average degree of the graph under consideration. model is O(davlogn).

Item Type:	Conference Paper
Publisher:	Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Additional Information:	Copyright of this article belongs to Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik.Germany.
Keywords:	Degeneracy;Cubicity;Boxicity;Crossing Number;Interval Graph;Intersection Graph;Poset Dimension;Comparability Graph;Random Graph; Average Degree
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	19 Mar 2013 09:15
Last Modified:	19 Mar 2013 09:15
URI:	http://eprints.iisc.ac.in/id/eprint/46036

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