Boxicity of Circular Arc Graphs

Bhowmick, Diptendu and Chandran, Sunil L (2011) Boxicity of Circular Arc Graphs. In: Graphs and Combinatorics, 27 (6). pp. 769-783.

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Abstract

A k-dimensional box is a Cartesian product R(1)x...xR(k) where each R(i) is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least pi(alpha-1/alpha) for some alpha is an element of N(>= 2), then box(G) <= alpha (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree Delta < [n(alpha-1)/2 alpha] for some alpha is an element of N(>= 2), then box(G) <= alpha. We also demonstrate a graph having box(G) > alpha but with Delta = n (alpha-1)/2 alpha + n/2 alpha(alpha+1) + (alpha+2). For a proper circular arc graph G, we show that if Delta < [n(alpha-1)/alpha] for some alpha is an element of N(>= 2), then box(G) <= alpha. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) <= r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k <= 3 arcs covers the circle, then box(G) <= 3 and if G admits a circular arc representation in which no family of k <= 4 arcs covers the circle, then box(G) <= 2. We also show that both these bounds are tight.

Item Type:	Journal Article
Publication:	Graphs and Combinatorics
Publisher:	Springer
Additional Information:	Copyright of this article belongs to Springer.
Keywords:	Boxicity;Circular arc graph;Minimum overlap set;Maximum circular cover number
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	04 Nov 2011 09:40
Last Modified:	04 Nov 2011 09:40
URI:	http://eprints.iisc.ac.in/id/eprint/41816

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