A set of vertices $W$ in a graph $G$ is called resolving if for any two distinct $x,yin V(G)$, there is $vin W$ such that ${rm dist}_G(v,x) eq{rm dist}_G(v,y)$, where ${rm dist}_G(u,v)$ denotes the length of a shortest path between $u$ and $v$ in the graph $G$. The metric dimension ${rm md}(G)$ of $G$ is the minimum cardinality of a resolving set. The Metric Dimension problem, i.e. deciding whether ${rm md}(G)le k$, is NP-complete even for interval graphs (Foucaud et al., 2017). We study Metric Dimension (for arbitrary graphs) from the lens of parameterized complexity. The problem parameterized by $k$ was proved to be $W[2]$-hard by Hartung and Nichterlein (2013) and we study the dual parameterization, i.e., the problem of whether ${rm md}(G)le n- k,$ where $n$ is the order of $G$. We prove that the dual parameterization admits (a) a kernel with at most $3k^4$ vertices and (b) an algorithm of runtime $O^*(4^{k+o(k)}).$ Hartung and Nichterlein (2013) also observed that Metric Dimension is fixed-parameter tractable when parameterized by the vertex cover number $vc(G)$ of the input graph. We complement this observation by showing that it does not admit a polynomial kernel even when parameterized by $vc(G) + k$. Our reduction also gives evidence for non-existence of polynomial Turing kernels.
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