Parameterized Complexity of Geodetic Set
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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, Geodetic Set asks whether there is a geodetic set of size at most k. We study the parameterized complexity of Geodetic Set with respect to structural parameters and show dichotomy results: We develop fixed-parameter algorithms with respect to the feedback edge number and with respect to the tree-depth. On the negative side, we prove that the problem is W[1]-hard with respect to the feedback vertex number and the path-width.
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