A d-dimensional box is a Cartesian product of d closed intervals on the real line. The boxicity of a graph is the minimum dimension d such that it is representable as the intersection graph of d-dimensional boxes. We give a short constructive proof that every graph with maximum degree D has boxicity at most 2D 2 . We also conjecture that the best upper bound is linear in D.

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An axis-parallel k-dimensional box is a Cartesian product R 1 × R 2 × · · · × R k where R i (for 1 ≤ i ≤ k) is a closed interval of the form [a i , b i ] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity 2 graphs has a 1 + 1 c log n approximation ratio for any constant c. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(∆ + 2) ln n⌉ dimension, where ∆ is the maximum degree of G. This algorithm implies that box(G) ≤ ⌈(∆ + 2) ln n⌉ for any graph G. Our bound is tight up to a factor of ln n. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree ∆, we show that for almost all graphs on n vertices, their boxicity is O(d av ln n) where d av is the average degree.

A piecewise linear simple curve in the plane made up of k + 1 line segments, each of which is either horizontal or vertical, with consecutive segments being of different orientation is called a k-bend path. Given a graph G, a collection of k-bend paths in which each path corresponds to a vertex in G and two paths have a common point if and only if the vertices corresponding to them are adjacent in G is called a B k -VPG representation of G. Similarly, a collection of k-bend paths each of which corresponds to a vertex in G is called an B k -EPG representation of G if any two paths have a line segment of non-zero length in common if and only if their corresponding vertices are adjacent in G. The VPG bend-number b v (G) of a graph G is the minimum k such that G has a B k -VPG representation. Similarly, the EPG bend-number b e (G) of a graph G is the minimum k such that G has a B k -EPG representation. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph then b v (G) ≤ 1 and b e (G) ≤ 2. These bounds are tight. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree to form a simple cycle, then it has a VPG-representation using only one type of 1-bend paths and an EPG-representation using only one type of 2-bend paths.

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