Boxicity of Circular Arc Graphs

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.