On the Structure of Hamiltonian Graphs with Small Independence Number
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929266034212864 |
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| author | Jedličková, Nikola Kratochvíl, Jan |
| author_facet | Jedličková, Nikola Kratochvíl, Jan |
| contents | 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. So far unpublished result of Jedličková and Kratochv\'ıl [arXiv:2309.09228] shows that for every integer $k$, Hamiltonian path and cycle are polynomial-time solvable in graphs of independence number bounded by $k$. As a companion structural result, 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 Hamiltonian path and cycle with no large hidden multiplicative constants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_03668 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Structure of Hamiltonian Graphs with Small Independence Number Jedličková, Nikola Kratochvíl, Jan Combinatorics Computational Complexity 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. So far unpublished result of Jedličková and Kratochv\'ıl [arXiv:2309.09228] shows that for every integer $k$, Hamiltonian path and cycle are polynomial-time solvable in graphs of independence number bounded by $k$. As a companion structural result, 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 Hamiltonian path and cycle with no large hidden multiplicative constants. |
| title | On the Structure of Hamiltonian Graphs with Small Independence Number |
| topic | Combinatorics Computational Complexity |
| url | https://arxiv.org/abs/2403.03668 |