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# Parameterized Complexity of Geodetic Set

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LIPIcs.IPEC.2020.20.pdf
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## Acknowledgements

We thank Lucia Draque Penso (Ulm University) for suggesting studying Geodetic Set from a view of parameterized complexity, and we thank André Nichterlein and Rolf Niedermeier (both TU Berlin) for helpful feedback and discussion. We are also grateful to an anonymous reviewer for suggesting that the ILP instances in Section 4 can be solved more efficiently.

## Cite As

Leon Kellerhals and Tomohiro Koana. Parameterized Complexity of Geodetic Set. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.IPEC.2020.20

## Abstract

A vertex set S of a graph G is geodetic if every vertex of G lies on a shortest path between two vertices in S. Given a graph G and k ∈ ℕ, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most k. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Parameterized complexity and exact algorithms
##### Keywords
• NP-hard graph problems
• Shortest paths
• Tree-likeness
• Parameter hierarchy
• Data reduction
• Integer linear programming

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