Simple Recognition of Halin Graphs and Their Generalizations

TL;DR

This paper introduces simple local reduction rules for recognizing Halin graphs in linear time, enabling efficient algorithms for various problems and extending to a broader class of polyhedral graphs.

Contribution

It presents new linear-time recognition algorithms for Halin graphs using local reduction rules, simplifying previous methods and extending to D3-reducible graphs.

Findings

Linear-time recognition of Halin graphs achieved

Reduction rules applicable to broader polyhedral graph class

D3-reducible graphs have bounded treewidth and Lombardi drawings

Abstract

We describe and implement two local reduction rules that can be used to recognize Halin graphs in linear time, avoiding the complicated planarity testing step of previous linear time Halin graph recognition algorithms. The same two rules can be used as the basis for linear-time algorithms for other algorithmic problems on Halin graphs, including decomposing these graphs into a tree and a cycle, finding a Hamiltonian cycle, or constructing a planar embedding. These reduction rules can also be used to recognize a broader class of polyhedral graphs. These graphs, which we call the D3-reducible graphs, are the dual graphs of the polyhedra formed by gluing pyramids together on their triangular faces; their treewidth is bounded, and they necessarily have Lombardi drawings.