Parameterized Vertex Deletion Problems for Hereditary Graph Classes with a Block Property

Bonnet, Édouard; Brettell, Nick; Kwon, O-joung; Marx, Dániel

doi:10.1007/978-3-662-53536-3_20

Cited by 11 publications

(9 citation statements)

References 27 publications

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“…The Feedback Vertex Set problem asks for a set S of at most k vertices such that G ´S is acyclic, or in other words, every block of G ´S is a single edge or a vertex. We consider generalizations where we allow the blocks to be some other type of small graph, such as triangles, small cycles, or small cliques; these generalizations were first studied in [5].…”

Section: Introductionmentioning

confidence: 99%

Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms

et al. 2019

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It has long been known that Feedback Vertex Set can be solved in time 2 Opw log wq n Op1q on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2 Opwq n Op1q , that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class P of graphs, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G´S has at most d vertices and is in P. Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d:• if P consists only of chordal graphs, then the problem can be solved in time 2 Opwd 2 q n Op1q ,• if P contains a graph with an induced cycle of length ě 4, then the problem is not solvable in time 2 opw log wq n Op1q even for fixed d " , unless the ETH fails.We also study a similar problem, called Bounded P-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if d is part of the input and P contains all chordal graphs, then it cannot be solved in time f pwqn opwq for some function f , unless the ETH fails.Bounded-size components. Using a similar technique, we can obtain analogous results for a simpler problem, which we call Bounded P-Component Vertex Deletion, where we want to remove at most k vertices such that each connected component of the resulting graph has at most d vertices and belongs to P. If we have only the size constraint (i.e., P contains every graph), then this problem is known as Component Order Connectivity [9]. Let P be a class of graphs.Bounded P-Component Vertex Deletion Parameter: d, w Input: A graph G of treewidth at most w, and positive integers d and k. Question: Is there a set S of at most k vertices in G such that each connected component of G´S has at most d vertices and is in P?Drange, Dregi, and van 't Hof [9] studied the parameterized complexity of a weighted variant of the Component Order Connectivity problem; their results imply, in particular, that Component Order Connectivity can be solved in time 2 Opk log dq n, but is W r1s-hard parameterized by only k or d. The corresponding edge-deletion problem, parameterized by treewidth, was studied by Enright and Meeks [10]. For general classes P, we prove results that are analogous to those for Bounded P-Block Vertex Deletion. Theorem 1.3. Let P be a class of graphs that is hereditary, recognizable in polynomial time, and consists of only chordal graphs. Then Bounded P-Component Vertex Deletion can be solved in time 2 Opwd 2 q k 2 n on graphs with n vertices and treewidth w.Theorem 1.4. Let P be a hereditary class of graphs that is polynomial-time recognizable. I...

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Section: Introductionmentioning

confidence: 99%

Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms

et al. 2019

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“…Hence we have t = 4. The problem Π 2 Vertex Deletion corresponds to Cactus Vertex Deletion which has a O * (26 k ) time FPT algorithm [3].…”

Section: Cliques or K T -Free Graph Subclassmentioning

confidence: 99%

Deletion to Scattered Graph Classes II -- Improved FPT Algorithms for Deletion to Pairs of Graph Classes

Jacob¹,

Majumdar²,

Raman³

2022

Preprint

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Let Π be a hereditary graph class. The problem of deletion to Π, takes as input a graph G and asks for a minimum number (or a fixed integer k) of vertices to be deleted from G so that the resulting graph belongs to Π. This is a well-studied problem in paradigms including approximation and parameterized complexity. This problem, for example, generalizes Vertex Cover, Feedback Vertex Set, Cluster Vertex Deletion, Perfect Vertex Deletion to name a few. The study of this problem in parameterized complexity has resulted in several powerful algorithmic techniques including iterative compression and important separators.Recently, the study of a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes, and the goal is to determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be FPT as long as the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. While this was shown using some black box theorems, faster algorithms were shown when each of the hereditary graph classes has a finite forbidden set.

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“…We write sG for the disjoint union of s copies as every independent set in a graph forms a solution. A restriction of the Max C-Block Graph problem was introduced and studied from a parameterized complexity perspective by Bonnet et al [4] as Bounded C-Block Vertex Deletion (so from the complementary perspective) where each block must in addition have bounded size.…”

Section: Introductionmentioning

confidence: 99%

Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

Paesani¹,

Paulusma²,

Rzążewski³

2021

Preprint

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We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for sP3-free graphs for every integer s ≥ 1. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs. ACM Subject Classification Mathematics of computing → Graph algorithmsKeywords and phrases Feedback vertex set, even cycle transversal, odd cactus, forest, block

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Parameterized Vertex Deletion Problems for Hereditary Graph Classes with a Block Property

Cited by 11 publications

References 27 publications

Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms

Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms

Deletion to Scattered Graph Classes II -- Improved FPT Algorithms for Deletion to Pairs of Graph Classes

Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

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