21 research outputs found

    Structural Parameterizations for Two Bounded Degree Problems Revisited

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    Parameterized Complexity of Vertex Splitting to Pathwidth at most 1

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    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 kk vertex explosions, where a vertex explosion replaces a vertex vv by deg(v)(v) degree-1 vertices, each incident to exactly one edge that was originally incident to vv. For POVE, we give an FPT algorithm with running time O(4km)O(4^k \cdot m) and an O(k2)O(k^2) kernel, thereby improving over the O(k6)O(k^6)-kernel by Ahmed et al. [GD 22] in a more general setting. Similarly, a vertex split replaces a vertex vv by two distinct vertices v1v_1 and v2v_2 and distributes the edges originally incident to vv arbitrarily to v1v_1 and v2v_2. 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 O((6k+12)km)O((6k+12)^k \cdot m). This answers an open question by Ahmed et al. [GD22]. Finally, we consider the problem Π\Pi Vertex Splitting (Π\Pi-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class Π\Pi using at most kk vertex splits. For graph classes Π\Pi that can be tested in monadic second-order graph logic (MSO2_2), we show that the problem Π\Pi-VS can be expressed as an MSO2_2 formula, resulting in an FPT algorithm for Π\Pi-VS parameterized by kk if Π\Pi additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions

    Approximately interpolating between uniformly and non-uniformly polynomial kernels

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    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 ρ-Treewidth Modulator problem (delete minimum number of vertices to ensure that treewidth is at most ρ) has a polynomial kernel of size k^g(ρ) 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

    Finding a Highly Connected Steiner Subgraph and its Applications

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    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

    A survey of parameterized algorithms and the complexity of edge modification

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    The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Slim Tree-Cut Width

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    Tree-cut width is a parameter that has been introduced as an attempt to obtain an analogue of treewidth for edge cuts. Unfortunately, in spite of its desirable structural properties, it turned out that tree-cut width falls short as an edge-cut based alternative to treewidth in algorithmic aspects. This has led to the very recent introduction of a simple edge-based parameter called edge-cut width [WG 2022], which has precisely the algorithmic applications one would expect from an analogue of treewidth for edge cuts, but does not have the desired structural properties. In this paper, we study a variant of tree-cut width obtained by changing the threshold for so-called thin nodes in tree-cut decompositions from 2 to 1. We show that this "slim tree-cut width" satisfies all the requirements of an edge-cut based analogue of treewidth, both structural and algorithmic, while being less restrictive than edge-cut width. Our results also include an alternative characterization of slim tree-cut width via an easy-to-use spanning-tree decomposition akin to the one used for edge-cut width, a characterization of slim tree-cut width in terms of forbidden immersions as well as an approximation algorithm for computing the parameter

    Parameterized Algorithms for Finding Large Sparse Subgraphs:Kernelization and Beyond

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    Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

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    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
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