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Deciding k-colourability of P_5-free graphs in polynomial time
[article]

2007
*
arXiv
*
pre-print

*In*contrast to this negative result, we show that determining whether or not a

*P_5*-

*free*

*graph*admits a

*k*-

*colouring*, for each fixed number

*of*

*colours*

*k*, can be done

*in*

*polynomial*

*time*. ... The problem

*of*computing the chromatic number

*of*a

*P_5*-

*free*

*graph*is known to be NP-hard. ... COROLLARY 1 Determining whether or not a P 5 -

*free*

*graph*can be

*coloured*with

*k*-

*colours*, for a fixed integer

*k*, can be

*decided*

*in*

*polynomial*

*time*. ...

##
###
Colouring (sP_1+P_5)-Free Graphs: a Mim-Width Perspective
[article]

2020
*
arXiv
*
pre-print

Then, as a consequence

arXiv:2004.05022v2
fatcat:h6itjk7nsvbzzi5cgkeuwrgvea
*of*our result, we obtain a new proof for the known result that for every fixed*k*≥ 1 and s≥ 0,*k*-*Colouring*is*polynomial*-*time*solvable for (sP_1+*P_5*)-*free**graphs*. ... We prove that the class*of*(K_t,sP_1+*P_5*)-*free**graphs*has bounded mim-width for every s≥ 0 and t≥ 1, and that there is a*polynomial*-*time*algorithm that, given a*graph**in*the class, computes a branch decomposition ... We now apply Lemma 5 to find that the mim-width*of*G − D is bounded and quickly computable. Let (T, δ) be a branch decomposition*of*G − D having mim-width*k*. ...##
###
List k-Colouring P_t-Free Graphs: a Mim-width Perspective
[article]

2021
*
arXiv
*
pre-print

Our result also generalizes the known result that for every

arXiv:2008.01590v2
fatcat:vkjvicp64bfvvlu2fssul32swi
*k*≥ 1 and s≥ 0, List*k*-*Colouring*is*polynomial*-*time*solvable for (sP_1+*P_5*)-*free**graphs*, which was proven for s=0 by Hoàng, Kamiński, Lozin, Sawada ... that List 3-*Colouring*is*polynomial*-*time*solvable for (K_1,s^1,P_t)-*free**graphs*for every t≥ 1 and s≥ 1. ...*In*particular, Hoàng et al. [20] proved that for every integer*k*≥ 1,*k*-*Colouring*is*polynomial*-*time*solvable for P 5 -*free**graphs*. Their proof is*in*fact a proof for List*k*-*Colouring*. ...##
###
Independent Feedback Vertex Set for P_5-free Graphs
[article]

2017
*
arXiv
*
pre-print

Tamura, Ito and Zhou proved that it is

arXiv:1707.09402v1
fatcat:6fpavkbv4vhslitxy6kwnw73ki
*polynomial*-*time*solvable for P_4-*free**graphs*. We show that it remains*polynomial*-*time*solvable for*P_5*-*free**graphs*. ... Finally,*in*line with our underlying research aim, we compare the complexity*of*Independent Feedback Vertex Set for H-*free**graphs*with the complexity*of*3-*Colouring*, Independent Odd Cycle Transversal and ... Feedback Vertex Set*in**polynomial**time*for P 5 -*free**graphs*. ...##
###
Connected Vertex Cover for $$(sP_1+P_5)$$(sP1+P5)-Free Graphs
[chapter]

2018
*
Lecture Notes in Computer Science
*

The Connected Vertex Cover problem is to

doi:10.1007/978-3-030-00256-5_23
fatcat:zdesmkjfxvf3fl3aayn5tymjyy
*decide*if a*graph*G has a vertex cover*of*size at most*k*that induces a connected subgraph*of*G. ... We prove that it is also*polynomial*-*time*solvable for (sP1 + P5)-*free**graphs*for every integer s ≥ 0. ... We also note that if Vertex Cover is*polynomial*-*time*solvable on H-*free**graphs*for some*graph*H, then it is*polynomial*-*time*solvable on (P 1 + H)-*free**graphs*. ...##
###
Connected Vertex Cover for (sP_1+P_5)-Free Graphs
[article]

2018
*
arXiv
*
pre-print

The Connected Vertex Cover problem is to

arXiv:1712.08362v3
fatcat:ues5vqajs5htxfcz3kmpozmsp4
*decide*if a*graph*G has a vertex cover*of*size at most*k*that induces a connected subgraph*of*G. ... We continue the search for tractable*graph*classes: we prove that it is also*polynomial*-*time*solvable for (sP_1+*P_5*)-*free**graphs*for every integer s≥ 0. ... We thank an anonymous reviewer*of*the conference version*of*our paper for helpful comments. ...##
###
On The Complexity of Matching Cut for Graphs of Bounded Radius and H-Free Graphs
[article]

2022
*
arXiv
*
pre-print

As a consequence

arXiv:2204.07129v3
fatcat:clztcrls6bgtzgkjxa6yhejrn4
*of*our result, we can solve Matching Cut*in**polynomial**time*for P_6-*free**graphs*, extending a recent result*of*Feghali for*P_5*-*free**graphs*. ... It is known that for an integer d, the corresponding decision problem Matching Cut is*polynomial*-*time*solvable for*graphs**of*diameter at most d if d≤ 2 and NP-complete if d≥ 3. ... We are now ready to prove the main result*of*this section. Theorem 4. Let H be a*graph*. If Matching Cut is*polynomial*-*time*solvable for H-*free**graphs*, then it is so for (H + P 3 )-*free**graphs*. Proof. ...##
###
Colouring (P_r+P_s)-Free Graphs
[article]

2018
*
arXiv
*
pre-print

We prove that List 3-

arXiv:1804.11091v1
fatcat:l7de742h2zbbxf7htcfnzwdwc4
*Colouring*is*polynomial*-*time*solvable for (P_2+*P_5*)-*free**graphs*and for (P_3+P_4)-*free**graphs*. ... The*k*-*Colouring*problem is to*decide*if the vertices*of*a*graph*can be*coloured*with at most*k**colours*for a fixed integer*k*such that no two adjacent vertices are*coloured*alike. ... We thank Karel Král for pointing out a mistake*in*a preliminary version*of*our paper. ...##
###
Independent Feedback Vertex Set for $$P_5$$P5-Free Graphs

2018
*
Algorithmica
*

Cover, which each require the desired set S

doi:10.1007/s00453-018-0474-x
fatcat:fky6jj6g4re3nfhv65svsydxxi
*of*size at most*k*to induce a connected*graph*, are also known to be NP-complete for line*graphs*and*graphs**of*arbitrarily large girth. ... Moreover, for these three problems the complexity has not yet been settled for H-*free**graphs*when H is a linear forest (see [13] for some partial results*in*this direction). ... Feedback Vertex Set*in**polynomial**time*for P 5 -*free**graphs*. ...##
###
Reconfiguration of vertex colouring and forbidden induced subgraphs
[article]

2022
*
arXiv
*
pre-print

Furthermore, we show that ℛ_k+1(G) is connected for every

arXiv:2206.09268v2
fatcat:xsxdkirhd5gilk7pj4z7m3soou
*k*-*colourable*(*P_5*, C_4)-*free**graph*G. ... The reconfiguration*graph**of*the*k*-*colourings*, denoted ℛ_k(G), is the*graph*whose vertices are the*k*-*colourings**of*G and two*colourings*are adjacent*in*ℛ_k(G) if they differ*in**colour*on exactly one vertex ... We note that this*colouring*can be found*in**polynomial**time*by optimally*colouring*each anticomponent*of*G [10] . Let A 1 , A 2 , . . . , A p be the anticomponents*of*G. ...##
###
Finding large H-colorable subgraphs in hereditary graph classes
[article]

2020
*
arXiv
*
pre-print

*in*

*polynomial*

*time*; ∙

*in*

*P_5*-

*free*

*graphs*

*in*

*time*n^O(ω(G)); ∙

*in*{P_6,1-subdivided claw}-

*free*

*graphs*

*in*

*time*n^O(ω(G)^3). ... We prove that for every fixed pattern

*graph*H without loops, Max Partial H-Coloring can be solved: ∙

*in*{

*P_5*,F}-

*free*

*graphs*

*in*

*polynomial*

*time*, whenever F is a threshold

*graph*; ∙

*in*{

*P_5*,bull}-

*free*

*graphs*...

*Deciding*

*k*-colorability

*of*P5 -

*free*

*graphs*

*in*

*polynomial*

*time*. Algorithmica, 57(1):74–81, 2010. [28] S. Huang. Improved complexity results on

*k*-coloring Pt -

*free*

*graphs*. Eur. J. ...

##
###
On Cycle Transversals and Their Connected Variants in the Absence of a Small Linear Forest
[article]

2019
*
arXiv
*
pre-print

We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved

arXiv:1908.00491v1
fatcat:xishk5douzbzjcwgmxnynoeecy
*in**polynomial**time*on (sP_1+P_3)-*free**graphs*... We complement these results by proving that Odd Cycle Transversal and Connected Odd Cycle Transversal are NP-complete on (P_2+*P_5*,P_6)-*free**graphs*. ... Independent Vertex Cover can be seen as 2-*Colouring*, with the additional restriction that one*of*the*colours*can be used at most*k**times*. This problem is*polynomial*-*time*solvable. ...##
###
Complexity of Coloring Graphs without Paths and Cycles
[article]

2013
*
arXiv
*
pre-print

have shown that

arXiv:1310.0340v1
fatcat:n5depkodjzfzjbahauiziha3vy
*k*-colorability*of**P_5*-*free**graphs*for*k*≥ 4 does not. ... We provide the full lists*of*forbidden induced subgraphs for*k*=3 and*k*=4. As an application, we obtain certifying*polynomial**time*algorithms for 3-coloring and 4-coloring (P_6,C_4)-*free**graphs*. ... (*In*fact the chromatic number*of*perfect*graphs*can also be computed*in**polynomial**time*[14] .) ...