The directed metric dimension of directed co-graphs
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A vertex w resolves two vertices u and v in a directed graph G if the distance from w to u is different to the distance from w to v. A set of vertices R is a resolving set for a directed graph G if for every pair of vertices u, v which are not in R there is at least one vertex in R that resolves u and v in G. The directed metric dimension of a directed graph G is the size of a minimum resolving set for G. The decision problem Directed Metric Dimension for a given directed graph G and a given number k is the question whether G has a resolving set of size at most k. In this paper, we study directed co-graphs. We introduce a linear time algorithm for computing a minimum resolving set for directed co-graphs and show that Directed Metric Dimension already is NP-complete for directed acyclic graphs.
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