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Deciding k-colourability of P_5-free graphs in polynomial time
[article]
2007
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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]
2018
Lecture Notes in Computer Science
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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]
2018
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2018
Algorithmica
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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]
2022
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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]
2013
arXiv
pre-print
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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