Abstract
We investigate the computational complexity of separating two distinct vertices s and z by vertex deletion in a temporal graph. In a temporal graph, the vertex set is fixed but the edges have (discrete) time labels. Since the corresponding Temporal (s, z)-Separation problem is NP-hard, it is natural to investigate whether relevant special cases exist that are computationally tractable. To this end, we study restrictions of the underlying (static) graph—there we observe polynomial-time solvability in the case of bounded treewidth—as well as restrictions concerning the “temporal evolution” along the time steps. Systematically studying partially novel concepts in this direction, we identify sharp borders between tractable and intractable cases.
Due to the space constraints, missing details and proofs (marked with \(\star \)) are deferred to a long version [9] of this paper, see https://arxiv.org/abs/1803.00882.
T. Fluschnik—Supported by the DFG, project DAMM (NI 369/13).
H. Molter—Supported by the DFG, project MATE (NI 369/17).
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Notes
- 1.
If this is not the case, then \(\varvec{E}\) can be sorted by ascending labels with bucketsort or mergesort in \(\mathcal {O}(\min \{\tau ,|\varvec{E}|\log |\varvec{E}|\})\) time.
- 2.
The vertex cover number of a graph is the smallest number of vertices such that each edges has at least one of these vertices as an endpoint.
- 3.
We refer to the long version [9] for details.
- 4.
The Kendall tau distance is a metric that counts the number of inversions between two total orderings; it is also known as “bubble sort distance”.
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Fluschnik, T., Molter, H., Niedermeier, R., Zschoche, P. (2018). Temporal Graph Classes: A View Through Temporal Separators. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_18
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