The crossing number of a graph is the least number of crossings over all drawings of the graph in the plane. Computing the crossing number of a given graph is NP-hard, but fixed-parameter tractable (FPT) with respect to the natural parameter. Two well-known variants of the problem are 2-layer crossing minimization and circular crossing minimization, where every vertex must lie on one of two layers, namely two parallel lines, or a circle, respectively. Both variants are NP-hard, but FPT with respect to the natural parameter. Recently, a local version of the crossing number has also received considerable attention. A graph is $k$-planar if it admits a drawing with at most $k$ crossings per edge. In contrast to the crossing number, recognizing $k$-planar graphs is NP-hard even if $k=1$. In this paper, we consider the two above variants in the local setting. The $k$-planar graphs that admit a straight-line drawing with vertices on two layers or on a circle are called 2-layer $k$-planar and outer $k$-planar graphs, respectively. We study the parameterized complexity of the two recognition problems with respect to $k$. For $k=0$, both problems can easily be solved in linear time. Two groups independently showed that outer 1-planar graphs can also be recognized in linear time [Hong et al., Algorithmica 2015; Auer et al., Algorithmica 2016]. One group asked whether outer 2-planar graphs can be recognized in polynomial time. Our main contribution consists of XP-algorithms for recognizing 2-layer $k$-planar graphs and outer $k$-planar graphs. We complement these results by showing that both recognition problems are XNLP-hard. This implies that both problems are W$[t]$-hard for every $t$ and that it is unlikely that they admit FPT-algorithms. On the other hand, we present an FPT-algorithm for recognizing 2-layer $k$-planar graphs where the order of the vertices on one layer is specified.

Given a graph G = (V,E), a set T of vertex pairs, and an integer k, Hitting Geodesic Intervals asks whether there is a set S ⊆ V of size at most k such that for each terminal pair {u,v} ∈ T, the set S intersects at least one shortest u-v path. Aravind and Saxena [WALCOM 2024] introduced this problem and showed several parameterized complexity results. In this paper, we extend the known results in both negative and positive directions and present sharp complexity contrasts with respect to structural graph parameters. We first show that the problem is NP-complete even on graphs with highly restricted shortest-path structures. More precisely, we show the NP-completeness on graphs obtained by adding a single vertex to a disjoint union of 5-vertex paths. By modifying the proof of this result, we also show the NP-completeness on graphs obtained from a path by adding one vertex and on graphs obtained from a disjoint union of triangles by adding one universal vertex. Furthermore, we show the NP-completeness on graphs of bandwidth 4 and maximum degree 5 by replacing the universal vertex in the last case with a long path. Under standard complexity assumptions, these negative results rule out fixed-parameter algorithms for most of the structural parameters studied in the literature (if the solution size k is not part of the parameter). We next present fixed-parameter algorithms parameterized by k plus modular-width and by k plus vertex integrity. The algorithm for the latter case does indeed solve a more general setting that includes the parameterization by the minimum vertex multiway-cut size of the terminal vertices. We show that this is tight in the sense that the problem parameterized by the minimum vertex multicut size of the terminal pairs is W[2]-complete. We then modify the proof of this intractability result and show that the problem is W[2]-complete parameterized by k even in the setting where T = binom(Q,2) for some Q ⊆ V.

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer $k$-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most $k$ other edges. Wood and Telle [New York J. Math., 2007] showed that every outer $k$-planar graph has treewidth at most $3k + 11$ using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer $1$-planar graphs is at most $3$, which is tight. In this paper, we improve the general upper bound to $1.5k + 2$ and give a tight bound of $4$ for $k = 2$. We also establish a lower bound: we show that, for every even $k$, there is an outer $k$-planar graph with treewidth $k+2$. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer $k$-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of $k + 2$ for the separation number of outer $k$-planar graphs, improving an upper bound of $2k + 3$ by Chaplick et al. [GD 2017]. We also consider outer min-$k$-planar graphs, a generalization of outer $k$-planar graphs, where we achieve smaller improvements.