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 ).
Highlights
Among the most fundamental problems in graph theory is the question whether a graph is Hamiltonian, i.e., contains a cycle of length n := |V |
Whitney [17] proved that every 4-connected maximal planar graph is Hamiltonian
There are numerous examples proving that 3-connected planar graphs are not necessarily Hamiltonian; even deciding whether a 3-connected 3-regular planar graph is Hamiltonian is NP-hard [10]
Summary
Among the most fundamental problems in graph theory is the question whether a graph is Hamiltonian, i.e., contains a cycle of length n := |V |. Much more is known for its preceding variants: Inspired by Tutte’s classic result, GouyouBeauchamps [11] showed that a Hamiltonian cycle in a 4-connected planar graph can be computed in polynomial time. The crux of this approach lies in the fact that the subgraphs arising from Tutte’s decomposition may overlap in an unbounded number of vertices and edges. We propose how to overcome this problem by extending the decomposition of Gao, Richter and Yu such that all arising subgraphs will be edge-disjoint This leads to the first polynomial-time algorithm that computes the special closed 2-walk of [8, 9], generalizing the previous results. A closed 2-walk of G can be computed in polynomial time such that x and y are visited exactly once and every vertex visited twice is contained in either a 2-separator or an internal 3-separator of G
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