3 research outputs found
Parameterized Complexity of Vertex Splitting to Pathwidth at most 1
Motivated by the planarization of 2-layered straight-line drawings, we
consider the problem of modifying a graph such that the resulting graph has
pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks
whether such a graph can be obtained using at most vertex explosions, where
a vertex explosion replaces a vertex by deg degree-1 vertices, each
incident to exactly one edge that was originally incident to . For POVE, we
give an FPT algorithm with running time and an
kernel, thereby improving over the -kernel by Ahmed et al. [GD 22] in a
more general setting. Similarly, a vertex split replaces a vertex by two
distinct vertices and and distributes the edges originally incident
to arbitrarily to and . Analogously to POVE, we define the
problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split
operation instead of vertex explosions. Here we obtain a linear kernel and an
algorithm with running time . This answers an open
question by Ahmed et al. [GD22].
Finally, we consider the problem Vertex Splitting (-VS), which
generalizes the problem POVS and asks whether a given graph can be turned into
a graph of a specific graph class using at most vertex splits. For
graph classes that can be tested in monadic second-order graph logic
(MSO), we show that the problem -VS can be expressed as an MSO
formula, resulting in an FPT algorithm for -VS parameterized by if
additionally has bounded treewidth. We obtain the same result for the
problem variant using vertex explosions
Parameterized Complexity of Simultaneous Planarity
Given input graphs , where each pair , with
shares the same graph , 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 . While SEFE is still open for the
case of two input graphs, the problem is NP-complete for [Schaefer,
JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We
show that SEFE is FPT with respect to plus the vertex cover number or the
feedback edge set number of the the union graph . Regarding the shared graph , we show that SEFE is NP-complete, even if
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 , 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