Boxicity of Halin graphs

A kk-dimensional box is the Cartesian product R1×R2×⋯×RkR1×R2×⋯×Rk where each RiRi is a closed interval on the real line. The boxicity of a graph GG, denoted as box(G) is the minimum integer kk such that GG is the intersection graph of a collection of kk-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 GG is a Halin graph that is not isomorphic to K4K4, then box(G)=2. In fact, we prove the stronger result that if GG is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G)=2 unless GG is isomorphic to K4K4 (in which case its boxicity is 1).