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Boxicity of Circular Arc Graphs

Authors

Abstract

A k -dimensional box is a Cartesian product R 1 × · · · × R 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 $${\pi(\frac{\alpha-1}{\alpha})}$$ for some $${\alpha\in\mathbb{N}_{\geq 2}}$$ , then box( G ) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree $${\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor}$$ for some $${\alpha \in \mathbb{N}_{\geq 2}}$$ , then box( G ) ≤ α . We also demonstrate a graph having box( G ) > α but with $${\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}$$ . For a proper circular arc graph G , we show that if $${\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor}$$ for some $${\alpha\in \mathbb{N}_{\geq 2}}$$ , then box( G ) ≤ α . 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.

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Journal

Graphs and CombinatoricsSpringer Journals

Published: Nov 1, 2011

DOI: 10.1007/s00373-010-1002-1

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