Small k-pyramids and the complexity of determining k

Motivated by the computational complexity of determining whether a graph is hamiltonian, we study under algorithmic aspects a class of polyhedra called k-pyramids, introduced in [Zamfirescu and Zamfirescu, Math. Nachr. 284 (2011) 1739–1747], and discuss related applications. We prove that determining whether a given graph is the 1-skeleton of a k-pyramid, and if so whether it is belted or not, can be done in polynomial time for k ≤ 3. The impact on hamiltonicity follows from the traceability of all 2-pyramids and non-belted 3-pyramids, and from the hamiltonicity of all non-belted 2-pyramids. The algorithm can also be used to determine the outcome for larger values of k, but the complexity increases exponentially with k. Lastly, we present applications of the algorithm, and improve the known bounds for the minimal cardinality of systems of bases called foundations in graph families with interesting properties concerning traceability and hamiltonicity.