Computing 2-Walks in Polynomial Time

Abstract

A 2-walk of a graph is a walk visiting every vertex at least once and at most twice. By generalizing decompositions of Tutte and Thomassen, Gao, Richter, and Yu proved that every 3-connected planar graph contains a closed 2-walk such that all vertices visited twice are contained in 3-separators. This seminal result generalizes Tutte’s theorem that every 4-connected planar graph is Hamiltonian, as well as Barnette’s theorem that every 3-connected planar graph has a spanning tree with maximum degree at most 3. The algorithmic challenge of finding such a closed 2-walk is to overcome big overlapping subgraphs in the decomposition, which are also inherent in Tutte’s and Thomassen’s decompositions. We solve this problem by extending the decomposition of Gao, Richter, and Yu in such a way that all pieces into which the graph is decomposed are edge-disjoint. This implies the first polynomial-time algorithm that computes the closed 2-walk just mentioned. Its running time is O ( n 3 ).