Abstract
A Hamiltonian path (cycle) in a graph is a path (cycle, respectively) which passes through all of its vertices. The problems of deciding the existence of a Hamiltonian cycle (path) in an input graph are well known to be NP-complete, and restricted classes of graphs which allow for their polynomial-time solutions are intensively investigated. Until very recently the complexity was open even for graphs of independence number at most 3. A so far unpublished result of Jedličková and Kratochvíl [arXiv:2309.09228] shows that for every integer k, the problems of deciding the existence of a Hamiltonian path and cycle are polynomial-time solvable in graphs of independence number bounded by k. As a companion structural result, in this paper, we determine explicit obstacles for the existence of a Hamiltonian path for small values of k, namely for graphs of independence number 2, 3, and 4. Identifying these obstacles in an input graph yields alternative polynomial-time algorithms for deciding the existence of a Hamiltonian path with no large hidden multiplicative constants.
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Jedličková, N., Kratochvíl, J. (2024). On the Structure of Hamiltonian Graphs with Small Independence Number. In: Rescigno, A.A., Vaccaro, U. (eds) Combinatorial Algorithms. IWOCA 2024. Lecture Notes in Computer Science, vol 14764. Springer, Cham. https://doi.org/10.1007/978-3-031-63021-7_14
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