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Hitting forbidden subgraphs in graphs of bounded treewidth [article]

Marek Cygan and Dániel Marx and Marcin Pilipczuk and Michał Pilipczuk
2014 arXiv   pre-print
We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph H and an input graph G, the task is to find a set X ⊆ V(G) of minimum size that hits all subgraphs  ...  In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors.  ...  Algorithms for Colorful H-Subgraph Hitting In this section we develop algorithmic upper bounds for Colorful H-Subgraph Hitting on graphs of bounded treewidth.  ... 
arXiv:1411.4184v1 fatcat:56y4ku7ocneadb7tluhtp3doee

Hitting Forbidden Subgraphs in Graphs of Bounded Treewidth [chapter]

Marek Cygan, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk
2014 Lecture Notes in Computer Science  
We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph H and an input graph G, the task is to find a set X ⊆ V (G) of minimum size that hits all subgraphs  ...  In the colorful variant of the problem, each vertex of G is precolored with some color from V (H) and we require to hit only H-subgraphs with matching colors.  ...  Algorithms for Colorful H-Subgraph Hitting In this section we develop algorithmic upper bounds for Colorful H-Subgraph Hitting on graphs of bounded treewidth.  ... 
doi:10.1007/978-3-662-44465-8_17 fatcat:ev7cwnlwqveadmur24qmsntehy

Hitting forbidden subgraphs in graphs of bounded treewidth

Marek Cygan, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk
2017 Information and Computation  
We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph H and an input graph G, we seek for the minimum size of a set X ⊆ V (G) that hits all subgraphs  ...  In the colorful variant of the problem, each vertex of G is precolored with some color from V (H) and we require to hit only H-subgraphs with matching colors.  ...  In the case of Colorful H-Subgraph Hitting, we obtain a tight bounds for the complexity of the treewidth parameterization.  ... 
doi:10.1016/j.ic.2017.04.009 fatcat:ajpyzikx2fbcldt6ltoqrpjsue

Optimality and tight results in parameterized complexity (Dagstuhl Seminar 14451)

Stefan Kratsch, Daniel Lokshtanov, Dániel Marx, Peter Rossmanith, Marc Herbstritt
2015 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 14451 "Optimality and tight results in parameterized complexity".  ...  The report contains the abstracts of the results presented at the seminar, as well as a collection of open problems stated at the seminar.  ...  Hitting Forbidden Subgraphs in Graphs of Bounded Treewidth O(t µ(H) ) • |V (G)|, but cannot be solved in time 2 o(t µ(H) ) • |V (G)| O Tutorial: Lower Bounds Based on the Exponential Time Hypothesis  ... 
doi:10.4230/dagrep.4.11.1 dblp:journals/dagstuhl-reports/KratschLMR14 fatcat:vwgmhttkgrf4pb34jpsb2rfvnm

Hitting forbidden induced subgraphs on bounded treewidth graphs [article]

Ignasi Sau, Uéverton S. Souza
2020 arXiv   pre-print
Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion  ...  We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version.  ...  This duality together with Lemma 4 1:10 Hitting forbidden induced subgraphs on bounded treewidth graphs yield the following key insight: in any graph G, after the removal of an optimal solution of I  ... 
arXiv:2004.08324v1 fatcat:6doqvwg6afcdvaf7ptfgx5qvwu

Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs

Ignasi Sau, Uéverton dos Santos Souza, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion  ...  We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version.  ...  The reason is that, in contrast to the subgraph version, when hitting induced subgraphs, edges and non-edges behave essentially in the same way when performing dynamic programming, as one has to keep track  ... 
doi:10.4230/lipics.mfcs.2020.82 dblp:conf/mfcs/SauS20 fatcat:lwuti2cppjdcxjhefajpssufye

Bounds on vertex colorings with restrictions on the union of color classes

N. R. Aravind, C. R. Subramanian
2011 Journal of Graph Theory  
Conclusions and Some Remarks We proved an upper bound of O(d k k+1−j ) on the (j, k)-chromatic number of graphs of maximum degree d and used it to obtain upper bounds for forbidden subgraph colorings.  ...  In general, graphs of treewidth at most k are minor-closed and hence it follows from the Graph Minor Theorem of Robertson and Seymour [17] that such graphs are characterized by a finite set of forbidden  ... 
doi:10.1002/jgt.20501 fatcat:2mbl2bcb3jepfe6chqf7efwlyy

Hitting forbidden minors: Approximation and Kernelization [article]

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, Saket Saurabh
2010 arXiv   pre-print
In an F-deletion problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden  ...  Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as Vertex Cover, Feedback Vertex Set and Diamond Hitting Set.  ...  The remark is not only true about graphs excluding K 1,t as an induced subgraph but also for any graph class G that can be characterized by either finite set of forbidden subgraphs or induced subgraphs  ... 
arXiv:1010.1365v1 fatcat:5abs6qzufja5dpj5gey52slyrq

Hitting forbidden minors: Approximation and Kernelization

Fedor V. Fomin, Daniel Lokshtanov, Neeldhara Misra, Geevarghese Philip, Saket Saurabh, Marc Herbstritt
2011 Symposium on Theoretical Aspects of Computer Science  
In an p-F -Deletion problem, we are asked whether a subset of at most k vertices can be deleted from a graph G such that the resulting graph does not contain as a minor any graph from the family F of forbidden  ...  We give • a linear vertex kernel on graphs excluding K 1,t as an induced subgraph, where t is a fixed integer. • an approximation algorithm achieving an approximation ratio of O(log 3/2 OPT ), where OPT  ...  We show that in this case the size of the hitting set returned by the algorithm can be bounded by O(k 2 log k).  ... 
doi:10.4230/lipics.stacs.2011.189 dblp:conf/stacs/FominLMPS11 fatcat:objrbhc4nbcndgaosbpvva2fhm

Isomorphism for Graphs of Bounded Feedback Vertex Set Number [chapter]

Stefan Kratsch, Pascal Schweitzer
2010 Lecture Notes in Computer Science  
For graphs of bounded feedback vertex set number the presented algorithm runs in time O(n 2 ), and is thereby the first fixed-parameter tractable Graph Isomorphism algorithm for a non-trivial parameter  ...  Central to the algorithm is a new technique consisting of an application of reduction rules that produce an isomorphism-invariant outcome, interleaved with the creation of increasingly large partial isomorphisms  ...  In fact the special case of vertex deletion can be solved by identifying all sets of at most d vertices that induce a forbidden subgraph and computing a minimum hitting set; using a current d-Hitting Set  ... 
doi:10.1007/978-3-642-13731-0_9 fatcat:ys6cnyk3qfaatfczzh24vkxlvm

Parameters Tied to Treewidth

Daniel J. Harvey, David R. Wood
2016 Journal of Graph Theory  
Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory.  ...  In a number of cases we also improve known bounds, provide simpler proofs, and show that the inequalities presented are tight. C 2016 Wiley Periodicals, Inc. J.  ...  the class of all graphs) has a finite set of forbidden minors.  ... 
doi:10.1002/jgt.22030 fatcat:t7225mp4qra5vlzpmwpi3cu6oi

Parameters Tied to Treewidth [article]

Daniel J. Harvey, David R. Wood
2016 arXiv   pre-print
Treewidth is a graph parameter of fundamental importance to algorithmic and structural graph theory.  ...  In a number of cases we also improve known bounds, provide simpler proofs and show that the inequalities presented are tight.  ...  the class of all graphs) has a finite set of forbidden minors.  ... 
arXiv:1312.3401v2 fatcat:ehinv26yrvbozfri36hmc4yus4

Edge Deletion to Restrict the Size of an Epidemic [article]

Ajinkya Gaikwad, Soumen Maity
2021 arXiv   pre-print
Given a graph G=(V,E), a set ℱ of forbidden subgraphs, we study ℱ-Free Edge Deletion, where the goal is to remove minimum number of edges such that the resulting graph does not contain any F∈ℱ as a subgraph  ...  A special case of particular interest is the situation in which ℱ is the set 𝒯_h+1 of all trees on h+1 vertices, so that we delete edges in order to obtain a graph in which every component contains at  ...  We are therefore interested in solving the following general problem: F -Free Edge Deletion Input: A graph G = (V, E), a set F of forbidden subgraphs and a positive integer k.  ... 
arXiv:2102.06068v1 fatcat:eflv7gnsinagvkf3wmfrnqiu7i

Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations

Bart M. P. Jansen, Astrid Pieterse, Michael Wagner
2018 European Symposium on Algorithms  
We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization.  ...  Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G − X.  ...  In Figure 1 (left), a construction of a graph of treewidth 2 together with a set Q is shown. In this graph, no optimal {K 2 }-deletion (Vertex Cover) breaks all graphs in Q.  ... 
doi:10.4230/lipics.esa.2018.48 dblp:conf/esa/JansenP18 fatcat:af55jmsen5hwpevlhjcb477i6u

Vertex Deletion Parameterized by Elimination Distance and Even Less [article]

Bart M. P. Jansen, Jari J. H. de Kroon, Michał Włodarczyk
2022 arXiv   pre-print
For the three mentioned vertex-deletion problems, and all problems which can be formulated as hitting a finite set of connected forbidden (a) minors or (b) (induced) subgraphs, we obtain FPT algorithms  ...  They are related to graph decompositions in which subgraphs that belong to a target class H (e.g., bipartite or planar) are considered simple.  ...  Hitting forbidden connected induced subgraphs In this section we deal with graph classes H defined by a finite family F of forbidden induced subgraphs.  ... 
arXiv:2103.09715v4 fatcat:vv3kluwdtjhqdhaflntr6msvdi
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