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The k-in-a-Path Problem for Claw-free Graphs

2010
*
Algorithmica
*

We show how to solve this

doi:10.1007/s00453-010-9468-z
fatcat:76mfyeqqvrfq3jo2ucurnkggmq
*problem**in*polynomial time on*claw*-*free**graphs*, when*k*is not part of*the*input but an arbitrarily fixed integer. ... Testing whether there is an induced*path**in**a**graph*spanning*k*given vertices is already NP-complete*in*general*graphs*when*k*=3. ... Acknowledgments We would like to thank*the*two anonymous referees*for*their helpful comments. ...##
###
THE k-IN-A-PATH PROBLEM FOR CLAW-FREE GRAPHS

2010
*
Symposium on Theoretical Aspects of Computer Science
*
unpublished

We show how to solve this

fatcat:6cjay7p3pbba7ag4haintyhfye
*problem**in*polynomial time on*claw*-*free**graphs*, when*k*is not part of*the*input but an arbitrarily fixed integer. ... Testing whether there is an induced*path**in**a**graph*spanning*k*given vertices is already NP-complete*in*general*graphs*when*k*= 3. ... Notice that any induced tree*in**a**claw*-*free**graph*is*in*fact an induced*path*. Observation 1.2.*The**k*-*in*-*a*-*Path*and*k*-*in*-*a*-Tree*problem*are equivalent*for**the*class of*claw*-*free**graphs*. Motivation. ...##
###
Induced Disjoint Paths in Claw-Free Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

Several related

doi:10.1007/978-3-642-33090-2_45
fatcat:dskh3ozjpbh5lmjcptcgjb6vte
*problems*, such as*the**k*-*in*-*a*-*Path**problem*, are proved to be fixed-parameter tractable*for**claw*-*free**graphs*as well. ... We show that this*problem*is fixed-parameter tractable*for**claw*-*free**graphs*when parameterized by*k*. ... We would like to thank two anonymous reviewers*for*their comments, which helped us improve*the*readability of our paper. ...##
###
Induced Disjoint Paths in Claw-Free Graphs
[article]

2014
*
arXiv
*
pre-print

Several related

arXiv:1202.4419v2
fatcat:gukgczgypzcttdbwqn4bxqswnm
*problems*, such as*the**k*-*in*-*a*-*Path**problem*, are proven to be fixed-parameter tractable*for**claw*-*free**graphs*as well. ... We show that this*problem*is fixed-parameter tractable*for**claw*-*free**graphs*when parameterized by*k*. ...*The**problems**k*-*in*-*a*-Cycle,*k*-*in*-*a*-*Path*, and*k*-*in*-*a*-Tree are fixed-parameter tractable*for**claw*-*free**graphs*when parameterized by*k*. Proof. First we consider*the**k*-*in*-*a*-Cycle*problem*. ...##
###
Induced Disjoint Paths in Claw-Free Graphs

2015
*
SIAM Journal on Discrete Mathematics
*

Several related

doi:10.1137/140963200
fatcat:t4rpbc6wgjfercr7dr6ogilzli
*problems*, such as*the**k*-*in*-*a*-*Path**problem*, are proved to be fixed-parameter tractable*for**claw*-*free**graphs*as well. ... We show that this*problem*is fixed-parameter tractable*for**claw*-*free**graphs*when parameterized by*k*. ... We would like to thank two anonymous reviewers*for*their comments, which helped us improve*the*readability of our paper. ...##
###
Packing 3-vertex paths in claw-free graphs and related topics

2011
*
Discrete Applied Mathematics
*

Let G be

doi:10.1016/j.dam.2010.05.001
fatcat:74tl7ztnxrdt3ccymbrplkpbay
*a*3-connected*claw*-*free**graph*, x*a*vertex*in*G, e = xy an edge*in*G, and P*a*3-vertex*path**in*G. ...*A**claw*is*a**graph*with four vertices and three edges incident to*the*same vertex.*A**graph*is*claw*-*free*if it has no induced subgraph isomorphic to*a**claw*. Our results include*the*following. ... (c12) We show that*the*-packing*problem**for**a**claw*-*free**graph*G can be reduced*in*polynomial time to that*for**a*special*claw*-*free**graph**K*(called*a*cactus) with v(*K*) ≤ v(G) (see 4.5 and 4.6). ...##
###
Deletion to Scattered Graph Classes II – Improved FPT Algorithms for Deletion to Pairs of Graph Classes
[article]

2022
*
arXiv
*
pre-print

*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. ... 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 ...

*Claw*-

*free*or Triangle-

*Free*

*graphs*We define

*the*

*problem*.

*Claw*-

*Free*-or-Triangle-

*Free*Deletion Input: An undirected

*graph*G = (V, E) and an integer

*k*. ...

##
###
Reconfiguring Independent Sets in Claw-Free Graphs
[chapter]

2014
*
Lecture Notes in Computer Science
*

We present

doi:10.1007/978-3-319-08404-6_8
fatcat:ktyhlpltubeefktjedu6b7kon4
*a*polynomial-time algorithm that, given two independent sets*in**a**claw*-*free**graph*G, decides whether one can be transformed into*the*other by*a*sequence of elementary steps. ... Each elementary step is to remove*a*vertex v from*the*current independent set S and to add*a*new vertex w (not*in*S) such that*the*result is again an independent set. ... This motivated*the*question asked*in*[11] whether*for*connected,*claw*-*free*and evenhole-*free**graph*G, TS*k*(G) is connected. ...##
###
Finding Induced Paths of Given Parity in Claw-Free Graphs

2010
*
Algorithmica
*

Our results imply that

doi:10.1007/s00453-010-9470-5
fatcat:emwqsmiedbfrnc4qd35kei5xki
*the**problem*of deciding if two specified vertices of*a**claw*-*free**graph*form an even pair, as well as*the**problem*of deciding if*a*given*claw*-*free**graph*has an even pair, can be solved ... Although all three*problems*are NP-complete*in*general, we show that they can be solved*in*O(n 5 ) time*for**the*class of*claw*-*free**graphs*. ...*The*authors would like to thank Jiří Fiala, Benjamin Lévêque, Luke Mathieson, Nicolas Trotignon and two anonymous reviewers*for*helpful suggestions. ...##
###
Finding Induced Paths of Given Parity in Claw-Free Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

Our results imply that

doi:10.1007/978-3-642-11409-0_30
fatcat:hstidpzhcjgvzka2vl6sd6yxma
*the**problem*of deciding if two specified vertices of*a**claw*-*free**graph*form an even pair, as well as*the**problem*of deciding if*a*given*claw*-*free**graph*has an even pair, can be solved ... Although all three*problems*are NP-complete*in*general, we show that they can be solved*in*O(n 5 ) time*for**the*class of*claw*-*free**graphs*. ...*The*authors would like to thank Jiří Fiala, Benjamin Lévêque, Luke Mathieson, Nicolas Trotignon and two anonymous reviewers*for*helpful suggestions. ...##
###
On linear and circular structure of (claw, net)-free graphs

2003
*
Discrete Applied Mathematics
*

We show also how one can use structural properties of (

doi:10.1016/s0166-218x(02)00571-1
fatcat:2pbmjgfx7zfj7hwy3cz2djmtlm
*claw*, net)-*free**graphs*to solve e ciently*the*domination, independent domination, and independent set*problems*on these*graphs*. ? ... Moreover, using LexBFS we present*a*linear time algorithm which,*for**a*given (*claw*, net)-*free**graph*, ÿnds either*a*dominating pair or an induced doubly dominating cycle. ...*In*[22] , Hempel and Kratsch gave linear time algorithms*for*all three*problems*on*the*class of*claw*-*free*AT-*free**graphs*. ...##
###
Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter
[article]

2021
*
arXiv
*
pre-print

*A*

*graph*G is H-

*free*if G does not contain H as an induced subgraph. It is known that

*for*all

*k*≥ 3,

*the*

*k*-Colouring

*problem*is NP-complete

*for*H-

*free*

*graphs*if H contains an induced

*claw*or cycle. ...

*For*

*a*fixed integer,

*the*

*k*-Colouring

*problem*is to decide if

*the*vertices of

*a*

*graph*can be coloured with at most

*k*colours

*for*an integer

*k*, such that no two adjacent vertices are coloured alike. ... What are

*the*complexities of 3-Colouring

*for*

*K*1 1,4 -

*free*

*graphs*of diameter 3 and

*for*

*K*2 1,3 -

*free*

*graphs*of diameter 3? Open

*Problem*4. ...

##
###
Intersection Graphs of Non-crossing Paths
[article]

2020
*
arXiv
*
pre-print

*A*direct consequence of our certifying algorithms is

*a*linear time algorithm certifying

*the*presence/absence of an induced

*claw*(K_1,3)

*in*

*a*chordal

*graph*. ...

*For*

*the*intersection

*graphs*of NC

*paths*of

*a*tree, we characterize

*the*minimum connected dominating sets (leading to

*a*linear time algorithm to compute one). ... This is relevant

*for*us because, as we will see

*in*Section 3,

*the*

*claw*-

*free*chordal

*graphs*are precisely

*the*NC-

*path*-tree

*graphs*. ...

##
###
On the Erdős-Gyárfás conjecture in claw-free graphs
[article]

2013
*
arXiv
*
pre-print

*In*this paper, we obtain some results on this question,

*in*particular

*for*cubic

*claw*-

*free*

*graphs*. ... Since this conjecture has proven to be far from reach, Hobbs asked if

*the*Erdős-Gyárfás conjecture holds

*in*

*claw*-

*free*

*graphs*. ...

*In*

*the*last section, we discuss on

*the*conjecture

*in*cubic

*claw*-

*free*

*graphs*. We also pose

*a*related conjecture and

*a*

*problem*. ...

##
###
From matchings to independent sets

2017
*
Discrete Applied Mathematics
*

It is well-known that finding

doi:10.1016/j.dam.2016.04.012
fatcat:vf4xsf7is5fxni5ullpnngptqq
*a*maximum matching*in*G is equivalent to finding*a*maximum independent set*in**the*line*graph*of G.*For*general*graphs*,*the*maximum independent set*problem*is NP-hard. ... We also review various techniques that may lead to efficient algorithms*for**the*maximum independent set*problem**in*restricted*graph*families, with*a*focus given to augmenting*graphs*and*graph*transformations ...*For**the*same reason,*paths*with an even number of vertices cannot be augmenting. Therefore, every connected*claw*-*free*augmenting*graph*is*a**path*P*k*with odd*k*. ...
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