Boxicity of Halin graphs

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Abstract

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). © 2008 Elsevier B.V. All rights reserved.

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Sunil Chandran, L., Francis, M. C., & Suresh, S. (2009). Boxicity of Halin graphs. Discrete Mathematics, 309(10), 3233–3237. https://doi.org/10.1016/j.disc.2008.09.037

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