Abstract
An edge Hamiltonian path of a graph is a permutation of its edge set where every pair of consecutive edges have a vertex in common. Unlike the seemingly related problem of finding an Eulerian walk, the edge Hamiltonian path is known to be a \(\mathsf {NP}\)-hard problem, even on fairly restricted classes of graphs. We introduce a natural optimization variant of the notion of an edge Hamiltonian path, which seeks the longest sequence of distinct edges with the property that every consecutive pair of them has a vertex in common. We call such a sequence of edges an edge-linked path, and study the parameterized complexity of the problem of finding edge-linked paths with at least k edges. We show that the problem is FPT when parameterized by k, and unlikely to admit a polynomial kernel even on connected graphs.
On the other hand, we show that the problem admits a Turing kernel of polynomial size. To the best of our knowledge, this is the first problem on general graphs to admit Turing kernels with adaptive oracles (for which a non-adaptive kernel is not known). We also design a single-exponential parameterized algorithm for the problem when parameterized by the treewidth of the input graph.
This work is supported by the European Research Council (ERC) via grant LOPPRE, reference 819416 and Norwegian Research Council via project MULTIVAL.
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Notes
- 1.
These instances typically have the property that the input instance is a Yes-instance if and only if one of these instances is a Yes-instance: therefore, the oracle can be applied to each instance in turn to solve the problem.
- 2.
Due to lack of space, the algorithm parameterized by treewidth and the arguments for the above-guarantee parameter are deferred to the full version of the paper.
- 3.
The proofs of results marked with \(\star \) are deferred to the full version of the paper.
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Misra, N., Panolan, F., Saurabh, S. (2019). On the Parameterized Complexity of Edge-Linked Paths. In: van Bevern, R., Kucherov, G. (eds) Computer Science – Theory and Applications. CSR 2019. Lecture Notes in Computer Science(), vol 11532. Springer, Cham. https://doi.org/10.1007/978-3-030-19955-5_25
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DOI: https://doi.org/10.1007/978-3-030-19955-5_25
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