Vertex Disjoint Path in Upward Planar Graphs

The $k$-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed $k$ when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter $k$) on this restricted class. Only recently, \cite{CMPP}.\ achieved a major breakthrough and answered the question positively. Despite the importance of this result (and the brilliance of their proof), it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has at least double exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller class of planar digraphs, the class of upward planar digraphs, a well studied class of planar graphs which can be drawn in a plane such that all edges are drawn upwards. We show that on the class of upward planar digraphs the problem (i) remains NP-complete and (ii) the problem is fixed-parameter tractable. While membership in FPT follows immediately from \cite{CMPP}'s general result, our algorithm has only single exponential parameter dependency compared to the double exponential parameter dependence for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in \cite{CMPP}.Comment: 14 page

Upward and Orthogonal Planarity are W[1]-hard Parameterized by Treewidth

Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both NP-complete, but XP when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known $n^{O(tw)}$-time algorithms cannot be improved to run in time $n^{o(tw)}$ unless ETH fails.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

On the Parameterized Complexity of Bend-Minimum Orthogonal Planarity

Computing planar orthogonal drawings with the minimum number of bends is one of the most relevant topics in Graph Drawing. The problem is known to be NP-hard, even when we want to test the existence of a rectilinear planar drawing, i.e., an orthogonal drawing without bends (Garg and Tamassia, 2001). From the parameterized complexity perspective, the problem is fixed-parameter tractable when parameterized by the sum of three parameters: the number of bends, the number of vertices of degree at most two, and the treewidth of the input graph (Di Giacomo et al., 2022). We improve this last result by showing that the problem remains fixed-parameter tractable when parameterized only by the number of vertices of degree at most two plus the number of bends. As a consequence, rectilinear planarity testing lies in \FPT~parameterized by the number of vertices of degree at most two.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Recognizing and Drawing IC-planar Graphs

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph $G$ with $n$ vertices, we present an $O(n)$-time algorithm that computes a straight-line drawing of $G$ in quadratic area, and an $O(n^3)$-time algorithm that computes a straight-line drawing of $G$ with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

Grid Recognition: Classical and Parameterized Computational Perspectives

Grid graphs, and, more generally, $k\times r$ grid graphs, form one of the most basic classes of geometric graphs. Over the past few decades, a large body of works studied the (in)tractability of various computational problems on grid graphs, which often yield substantially faster algorithms than general graphs. Unfortunately, the recognition of a grid graph is particularly hard -- it was shown to be NP-hard even on trees of pathwidth 3 already in 1987. Yet, in this paper, we provide several positive results in this regard in the framework of parameterized complexity (additionally, we present new and complementary hardness results). Specifically, our contribution is threefold. First, we show that the problem is fixed-parameter tractable (FPT) parameterized by $k+\mathsf {mcc}$ where $\mathsf{mcc}$ is the maximum size of a connected component of $G$. This also implies that the problem is FPT parameterized by $\mathtt{td}+k$ where $\mathtt{td}$ is the treedepth of $G$ (to be compared with the hardness for pathwidth 2 where $k=3$). Further, we derive as a corollary that strip packing is FPT with respect to the height of the strip plus the maximum of the dimensions of the packed rectangles, which was previously only known to be in XP. Second, we present a new parameterization, denoted $a_G$, relating graph distance to geometric distance, which may be of independent interest. We show that the problem is para-NP-hard parameterized by $a_G$, but FPT parameterized by $a_G$ on trees, as well as FPT parameterized by $k+a_G$. Third, we show that the recognition of $k\times r$ grid graphs is NP-hard on graphs of pathwidth 2 where $k=3$. Moreover, when $k$ and $r$ are unrestricted, we show that the problem is NP-hard on trees of pathwidth 2, but trivially solvable in polynomial time on graphs of pathwidth 1

Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions