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

Chính T. Hoàng, Marcin Kamiński, Vadim Lozin, J. Sawada, X. Shu
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.  ... 
arXiv:cs/0702043v1 fatcat:sm4466tmyfdylbp3g4gwxhgcqy

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

Nick Brettell, Jake Horsfield, Daniel Paulusma
2020 arXiv   pre-print
Then, as a consequence 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.  ... 
arXiv:2004.05022v2 fatcat:h6itjk7nsvbzzi5cgkeuwrgvea

List k-Colouring P_t-Free Graphs: a Mim-width Perspective [article]

Nick Brettell, Jake Horsfield, Andrea Munaro, Daniel Paulusma
2021 arXiv   pre-print
Our result also generalizes the known result that for every 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.  ... 
arXiv:2008.01590v2 fatcat:vkjvicp64bfvvlu2fssul32swi

Independent Feedback Vertex Set for P_5-free Graphs [article]

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, Daniel Paulusma
2017 arXiv   pre-print
Tamura, Ito and Zhou proved that it is 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.  ... 
arXiv:1707.09402v1 fatcat:6fpavkbv4vhslitxy6kwnw73ki

Connected Vertex Cover for $$(sP_1+P_5)$$(sP1+P5)-Free Graphs [chapter]

Matthew Johnson, Giacomo Paesani, Daniël Paulusma
2018 Lecture Notes in Computer Science  
The Connected Vertex Cover problem is to 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.  ... 
doi:10.1007/978-3-030-00256-5_23 fatcat:zdesmkjfxvf3fl3aayn5tymjyy

Connected Vertex Cover for (sP_1+P_5)-Free Graphs [article]

Matthew Johnson and Giacomo Paesani and Daniel Paulusma
2018 arXiv   pre-print
The Connected Vertex Cover problem is to 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.  ... 
arXiv:1712.08362v3 fatcat:ues5vqajs5htxfcz3kmpozmsp4

On The Complexity of Matching Cut for Graphs of Bounded Radius and H-Free Graphs [article]

Felicia Lucke and Daniël Paulusma and Bernard Ries
2022 arXiv   pre-print
As a consequence 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.  ... 
arXiv:2204.07129v3 fatcat:clztcrls6bgtzgkjxa6yhejrn4

Colouring (P_r+P_s)-Free Graphs [article]

Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, Veronika Slívová
2018 arXiv   pre-print
We prove that List 3-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.  ... 
arXiv:1804.11091v1 fatcat:l7de742h2zbbxf7htcfnzwdwc4

Independent Feedback Vertex Set for $$P_5$$P5-Free Graphs

Marthe Bonamy, Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, Daniël Paulusma
2018 Algorithmica  
Cover, which each require the desired set S 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.  ... 
doi:10.1007/s00453-018-0474-x fatcat:fky6jj6g4re3nfhv65svsydxxi

Reconfiguration of vertex colouring and forbidden induced subgraphs [article]

Manoj Belavadi, Kathie Cameron, Owen Merkel
2022 arXiv   pre-print
Furthermore, we show that ℛ_k+1(G) is connected for every 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.  ... 
arXiv:2206.09268v2 fatcat:xsxdkirhd5gilk7pj4z7m3soou

Finding large H-colorable subgraphs in hereditary graph classes [article]

Maria Chudnovsky, Jason King, Michał Pilipczuk, Paweł Rzążewski, Sophie Spirkl
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.  ... 
arXiv:2004.09425v2 fatcat:xjposflbmbafhhgqksipvmw3sm

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

Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, Giacomo Paesani, Daniël Paulusma, Paweł Rzążewski
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 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.  ... 
arXiv:1908.00491v1 fatcat:xishk5douzbzjcwgmxnynoeecy

Complexity of Coloring Graphs without Paths and Cycles [article]

Pavol Hell, Shenwei Huang
2013 arXiv   pre-print
have shown that 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] .)  ... 
arXiv:1310.0340v1 fatcat:n5depkodjzfzjbahauiziha3vy