9 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
Finding a Highly Connected Steiner Subgraph and its Applications
Given a (connected) undirected graph G, a set X ? V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ? X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness).
Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph
On Approximate Compressions for Connected Minor-Hitting Sets
In the Connected ?-Deletion problem, ? is a fixed finite family of graphs and the objective is to compute a minimum set of vertices (or a vertex set of size at most k for some given k) such that (a) this set induces a connected subgraph of the given graph and (b) deleting this set results in a graph which excludes every F ? ? as a minor. In the area of kernelization, this problem is well known to exclude a polynomial kernel subject to standard complexity hypotheses even in very special cases such as ? = K?, i.e., Connected Vertex Cover.
In this work, we give a (2+?)-approximate polynomial compression for the Connected ?-Deletion problem when ? contains at least one planar graph. This is the first approximate polynomial compression result for this generic problem. As a corollary, we obtain the first approximate polynomial compression result for the special case of Connected ?-Treewidth Deletion
Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size
In the ?-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G-S contains no graph from the finite family ? as a minor. It is known that whenever ? contains at least one planar graph, then ?-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size k^{?(1)} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size.
We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of ?-Minor-Free Deletion for the family ? = {K?, K_{2,3}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with ?(k?) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has ?(k?) vertices and edges
An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and
Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B,
96(4):514--528, 2006]. Motivated from recent development on graph modification
problems regarding classes of graphs of bounded treewidth or pathwidth, we
study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex
Deletion). In the LRW1-Vertex Deletion problem, given an -vertex graph
and a positive integer , we want to decide whether there is a set of at most
vertices whose removal turns into a graph of linear rankwidth at most
and find such a vertex set if one exists. While the meta-theorem of
Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved
in time for some function , it is not clear whether this
problem allows a running time with a modest exponential function.
We first establish that LRW1-Vertex Deletion can be solved in time . The major obstacle to this end is how to handle a long
induced cycle as an obstruction. To fix this issue, we define necklace graphs
and investigate their structural properties. Later, we reduce the polynomial
factor by refining the trivial branching step based on a cliquewidth expression
of a graph, and obtain an algorithm that runs in time . We also prove that the running time cannot be improved to under the Exponential Time Hypothesis assumption. Lastly,
we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201
Approximately interpolating between uniformly and non-uniformly polynomial kernels
The problem of computing a minimum set of vertices intersecting a finite set of forbidden minors in a given graph is a fundamental graph problem in the area of kernelization with numerous well-studied special cases. A major breakthrough in this line of research was made by Fomin et al. [FOCS 2012], who showed that the Ï-Treewidth Modulator problem (delete minimum number of vertices to ensure that treewidth is at most Ï) has a polynomial kernel of size k^g(Ï) for some function g. A second standout result in this line is that of Giannapoulou et al. [ACM TALG 2017], who obtained an f(η)k