Boxicity of series-parallel graphs

The three well-known graph classes, planar graphs(P), series-parallel graphs (SP) and outer planar graphs(OP) satisfy the following proper inclusion relation: OP ⊂ SP ⊂ P . It is known that box(G) ≤ 3 if G ∈ P and box(G) ≤ 2 if G ∈ OP . Thus it is interesting to decide whether the maximum possible value of the boxicity of series-parallel graphs is 2 or 3. In this paper we construct a series-parallel graph with boxicity 3, thus resolving this question. Recently Chandran and Sivadasan [3] showed that for any G, box(G) ≤ treewidth(G) + 2. They conjecture that for any k, there exists a k-tree with boxicity k + 1. (This would show that their upper bound is tight but for an additive factor of 1, since the treewidth of any k-tree equals k.) The series-parallel graph we construct in this paper is a 2-tree with boxicity 3 and is thus a first step towards proving their conjecture.