Abstract
In this paper, we consider three hitting problems on a disk intersection graph: Triangle Hitting Set, Feedback Vertex Set, and Odd Cycle Transversal. Given a disk intersection graph G, our goal is to compute a set of vertices hitting all triangles, all cycles, or all odd cycles, respectively. Our algorithms run in time \(2^{\tilde{O}({k}^{4/5})}n^{O(1)}\), \(2^{\tilde{O}({k}^{9/10})}n^{O(1)}\), and \(2^{\tilde{O}({k}^{19/20})}n^{O(1)}\), respectively, where n denotes the number of vertices of G. These do not require a geometric representation of a disk graph. If a geometric representation of a disk graph is given as input, we can solve these problems more efficiently. In this way, we improve the algorithms for those three problem by Lokshtanov et al. [SODA 2022].
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No.RS-2023-00209069).
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Notes
- 1.
The definition of the treewidth can be found in Sect. 2.
- 2.
We can find a hitting set of size at most 3k as follows: Find a triangle, and add all its vertices to a triangle hitting set. Then remove all its vertices from the graph.
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An, S., Cho, K., Oh, E. (2023). Faster Algorithms for Cycle Hitting Problems on Disk Graphs. In: Morin, P., Suri, S. (eds) Algorithms and Data Structures. WADS 2023. Lecture Notes in Computer Science, vol 14079. Springer, Cham. https://doi.org/10.1007/978-3-031-38906-1_3
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