Boxicity of Halin graphs

A k -dimensional box is the Cartesian product R 1 × R 2 × ⋯ × 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 is the intersection graph of a collection of k -dimensional boxes. 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 that is not isomorphic to K 4 , then box ( G ) = 2 . In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box ( G ) = 2 unless G is isomorphic to K 4 (in which case its boxicity is 1).