## Small $$k$$-pyramids and the complexity of determining $$k$$.(English)Zbl 1320.05130

Summary: 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 [C. T. Zamfirescu and T. I. Zamfirescu, Math. Nachr. 284, No. 13, 1739–1747 (2011; Zbl 1236.05121)], 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 $$k \leq 3$$. 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.

### MSC:

 05C85 Graph algorithms (graph-theoretic aspects) 05C45 Eulerian and Hamiltonian graphs

### Keywords:

pyramid; prism; Halin graph; Hamiltonian

Zbl 1236.05121
Full Text:

### References:

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