On the Structure of Hamiltonian Graphs with Small Independence Number
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. So far unpublished result of Jedličková and Kratochvíl [arXiv:2309.09228] shows that for every integer $k$, Hamiltonian path and cycle are polynomial-time solvable in graphs of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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2024
- DOI:
- 10.48550/arXiv.2403.03668
- arXiv:
- arXiv:2403.03668
- Bibcode:
- 2024arXiv240303668J
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Computational Complexity
- E-Print:
- arXiv admin note: text overlap with arXiv:2309.09228