Embeddings of cubic Halin graphs: Genus distributions

Abstract

We derive an O(n²)-time algorithm for calculating the genus distribution of a given 3-regular Halin graph G; that is, we calculate the sequence of numbers g₀(G), g₁(G), g₂(G), … on the respective orientable surfaces S₀, S₁, S₂, …. Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinant-strands reassembly process that fits pieces together three-at-a-vertex. Key algorithmic features are reassembly along a post-order traversal, with just-in-time dynamic assignment of roots for quadrangular pieces encountered along the tour.