Abstract
The metric dimension has been introduced independently by Harary, Melter [11] and Slater [15] in 1975 to identify vertices of a graph G using its distances to a subset of vertices of G. A resolving set X of a graph G is a subset of vertices such that, for every pair (u, v) of vertices of G, there is a vertex x in X such that the distance between x and u and the distance between x and v are distinct. The metric dimension of the graph is the minimum size of a resolving set. Computing the metric dimension of a graph is NP-hard even on split graphs and interval graphs. Bonnet and Purohit [2] proved that the metric dimension problem is W[1]-hard parameterized by treewidth. Li and Pilipczuk strengthened this result by showing that it is NP-hard for graphs of treewidth 24 in [14]. In this article, we prove that metric dimension is FPT parameterized by treewidth in chordal graphs.
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Notes
- 1.
The addition of a single edge in a graph might modify the metric dimension by \(\varOmega (n)\), see e.g. [7].
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Bousquet, N., Deschamps, Q., Parreau, A. (2023). Metric Dimension Parameterized by Treewidth in Chordal Graphs. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_10
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