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Abstract. The metric dimension of a graph G is the size of a smallest subset L ⊆ V (G) such that for any x, y ∈ V (G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomialtime solvable on trees, and to have a log n-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

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Paths P1,. .. , P k in a graph G = (V, E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si, ti) contains k mutually induced paths Pi such that each Pi connects si and ti. This is a classical graph problem that is NP-complete even for k = 2. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for ATfree graphs, we prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs, for which such a result only holds if k is fixed. As a consequence of our main result, the problem of deciding if a given ATfree graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |VH |, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems kin a Path and kin a Tree are polynomialtime solvable on AT-free graphs even if k is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning k given vertices.

The well-known DISJOINT PATHS problem takes as input a graph G and a set of k pairs of terminals in G, and the task is to decide whether there exists a collection of k pairwise vertex-disjoint paths in G such that the vertices in each terminal pair are connected to each other by one of the paths. This problem is known to be NP-complete, even when restricted to planar graphs or interval graphs. Moreover, although the problem is fixed-parameter tractable when parameterized by k due to a celebrated result by Robertson and Seymour, it is known not to admit a polynomial kernel unless NP ⊆ coNP/poly. We prove that DISJOINT PATHS remains NP-complete on split graphs, and show that the problem admits a kernel with O(k 2 ) vertices when restricted to this graph class. We furthermore prove that, on split graphs, the edge-disjoint variant of the problem is also NP-complete and admits a kernel with O(k 3 ) vertices. To the best of our knowledge, our kernelization results are the first non-trivial kernelization results for these problems on graph classes.