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Hitting forbidden subgraphs in graphs of bounded treewidth
[article]

2014
*
arXiv
*
pre-print

We study the complexity

arXiv:1411.4184v1
fatcat:56y4ku7ocneadb7tluhtp3doee
*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*. ...##
###
Hitting Forbidden Subgraphs in Graphs of Bounded Treewidth
[chapter]

2014
*
Lecture Notes in Computer Science
*

We study the complexity

doi:10.1007/978-3-662-44465-8_17
fatcat:ev7cwnlwqveadmur24qmsntehy
*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*. ...##
###
Hitting forbidden subgraphs in graphs of bounded treewidth

2017
*
Information and Computation
*

We study the complexity

doi:10.1016/j.ic.2017.04.009
fatcat:ajpyzikx2fbcldt6ltoqrpjsue
*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. ...##
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Optimality and tight results in parameterized complexity (Dagstuhl Seminar 14451)

2015
*
Dagstuhl Reports
*

This report documents the program and the outcomes

doi:10.4230/dagrep.4.11.1
dblp:journals/dagstuhl-reports/KratschLMR14
fatcat:vwgmhttkgrf4pb34jpsb2rfvnm
*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 ...##
###
Hitting forbidden induced subgraphs on bounded treewidth graphs
[article]

2020
*
arXiv
*
pre-print

Motivated by previous work about

arXiv:2004.08324v1
fatcat:6doqvwg6afcdvaf7ptfgx5qvwu
*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 ...##
###
Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

Motivated by previous work about

doi:10.4230/lipics.mfcs.2020.82
dblp:conf/mfcs/SauS20
fatcat:lwuti2cppjdcxjhefajpssufye
*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 ...##
###
Bounds on vertex colorings with restrictions on the union of color classes

2011
*
Journal of Graph Theory
*

Conclusions and Some Remarks We proved an upper

doi:10.1002/jgt.20501
fatcat:2mbl2bcb3jepfe6chqf7efwlyy
*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*...##
###
Hitting forbidden minors: Approximation and Kernelization
[article]

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*...

##
###
Hitting forbidden minors: Approximation and Kernelization

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). ...

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

2010
*
Lecture Notes in Computer Science
*

For

doi:10.1007/978-3-642-13731-0_9
fatcat:ys6cnyk3qfaatfczzh24vkxlvm
*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 ...##
###
Parameters Tied to Treewidth

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. ...

##
###
Parameters Tied to Treewidth
[article]

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. ...

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

2021
*
arXiv
*
pre-print

Given a

arXiv:2102.06068v1
fatcat:eflv7gnsinagvkf3wmfrnqiu7i
*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. ...##
###
Polynomial Kernels for Hitting Forbidden Minors under Structural Parameterizations

2018
*
European Symposium on Algorithms
*

We investigate polynomial-time preprocessing for the problem

doi:10.4230/lipics.esa.2018.48
dblp:conf/esa/JansenP18
fatcat:af55jmsen5hwpevlhjcb477i6u
*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. ...##
###
Vertex Deletion Parameterized by Elimination Distance and Even Less
[article]

2022
*
arXiv
*
pre-print

For the three mentioned vertex-deletion problems, and all problems which can be formulated as

arXiv:2103.09715v4
fatcat:vv3kluwdtjhqdhaflntr6msvdi
*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*. ...
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