Boxicity of Halin Graphs

TL;DR

This paper establishes that all Halin graphs, except K_4, have a boxicity of 2, meaning they can be represented as intersections of 2-dimensional boxes, which advances understanding of their geometric representations.

Contribution

The paper proves that non-K_4 Halin graphs and similar planar graphs have a boxicity of 2, providing a precise characterization of their geometric intersection properties.

Findings

Halin graphs (not K_4) have boxicity 2

K_4 has boxicity 1

Planar graphs formed by connecting leaves in a cycle also have boxicity 2

Abstract

A k-dimensional box is the Cartesian product R_1 x R_2 x ... x 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).