Adiga, Abhijin and Chandran, Sunil L (2010) Cubicity of Interval Graphs and the Claw Number. In: Journal of Graph Theory, 65 (4). pp. 323333.

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Abstract
Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A bdimensional cube is a Cartesian product l(1) x l(2) x ... x l(b), where each l(i) is a closed interval of unit length on the real line. The cub/city of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel bdimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real linei.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number psi(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least log(2) psi(G)]. In this article, we show that for an interval graph G log(2) psi(G)]<= cub(G)<=log(2) psi(G)]+2. It is not clear whether the upper bound of log(2) psi(G)]+2 is tight: till now we are unable to find any interval graph with cub(G)> (log(2)psi(G)]. We also show that for an interval graph G, cub(G) <= log(2) alpha], where alpha is the independence number of G. Therefore, in the special case of psi(G)=alpha, cub(G) is exactly log(2) alpha(2)]. The concept of cubicity can be generalized by considering boxes instead of cubes. A bdimensional box is a Cartesian product l(1) x l(2) x ... x l(b), where each I is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of kdimensional boxes. It is clear that box(G)<= cub(G). From the above result, it follows that for any graph G, cub(G) <= box(G)log(2) alpha]. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323333, 2010
Item Type:  Journal Article 

Publication:  Journal of Graph Theory 
Publisher:  John Wiley & Sons 
Additional Information:  Copyright of this article belongs to John Wiley & Sons. 
Keywords:  cubicity; boxicity;interval graphs;indifference graphs;claw number 
Department/Centre:  Division of Electrical Sciences > Computer Science & Automation 
Date Deposited:  08 Dec 2010 11:19 
Last Modified:  29 Feb 2012 07:18 
URI:  http://eprints.iisc.ac.in/id/eprint/34313 
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