Parameterized Complexity of Simultaneous Planarity

Given $k$ input graphs $G_1, \dots ,G_k$, where each pair $G_i$, $G_j$ with $i \neq j$ shares the same graph $G$, the problem Simultaneous Embedding With Fixed Edges (SEFE) asks whether there exists a planar drawing for each input graph such that all drawings coincide on $G$. While SEFE is still open for the case of two input graphs, the problem is NP-complete for $k \geq 3$ [Schaefer, JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We show that SEFE is FPT with respect to $k$ plus the vertex cover number or the feedback edge set number of the the union graph $G^\cup = G_1 \cup \dots \cup G_k$. Regarding the shared graph $G$, we show that SEFE is NP-complete, even if $G$ is a tree with maximum degree 4. Together with a known NP-hardness reduction [Angelini et al., TCS 15], this allows us to conclude that several parameters of $G$, including the maximum degree, the maximum number of degree-1 neighbors, the vertex cover number, and the number of cutvertices are intractable. We also settle the tractability of all pairs of these parameters. We give FPT algorithms for the vertex cover number plus either of the first two parameters and for the number of cutvertices plus the maximum degree, whereas we prove all remaining combinations to be intractable.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Experimental Comparison of PC-Trees and PQ-Trees

PQ-trees and PC-trees are data structures that represent sets of linear and circular orders, respectively, subject to constraints that specific subsets of elements have to be consecutive. While equivalent to each other, PC-trees are conceptually much simpler than PQ-trees; updating a PC-tree so that a set of elements becomes consecutive requires only a single operation, whereas PQ-trees use an update procedure that is described in terms of nine transformation templates that have to be recursively matched and applied. Despite these theoretical advantages, to date no practical PC-tree implementation is available. This might be due to the original description by Hsu and McConnell [Hsu et al., 2003] in some places only sketching the details of the implementation. In this paper, we describe two alternative implementations of PC-trees. For the first one, we follow the approach by Hsu and McConnell, filling in the necessary details and also proposing improvements on the original algorithm. For the second one, we use a different technique for efficiently representing the tree using a Union-Find data structure. In an extensive experimental evaluation we compare our implementations to a variety of other implementations of PQ-trees that are available on the web as part of academic and other software libraries. Our results show that both PC-tree implementations beat their closest fully correct competitor, the PQ-tree implementation from the OGDF library [Markus Chimani et al., 2014; Leipert, 1997], by a factor of 2 to 4, showing that PC-trees are not only conceptually simpler but also fast in practice. Moreover, we find the Union-Find-based implementation, while having a slightly worse asymptotic runtime, to be twice as fast as the one based on the description by Hsu and McConnell