Maximum weight stable set in (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msub></mml:math>, bull)-free graphs and (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml2" display="inline" overflow="scroll" altimg="si2.gif"><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mm

Maffray, Frédéric; Pastor, Lucas

doi:10.1016/j.disc.2017.10.004

Cited by 6 publications

(3 citation statements)

References 27 publications

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“…In this paper, we show that for (S 1,2,4 ,triangle)-free graphs, MWIS can be solved in polynomial time. This generalizes the polynomial-time result for MWIS on (P 7 ,triangle)-free graphs [7] (which was extended by Maffray and Pastor [15] showing that MWIS can be solved in polynomial time for (P 7 ,bull)-free graphs; in the same paper, they also showed that MWIS can be solved in polynomial time for (S 1,2,3 ,bull)-free graphs).…”

Section: Introductionsupporting

confidence: 80%

Maximum Weight Independent Sets for (P7,triangle)-free graphs in polynomial time

Brandstädt

Mosca

2018

Discrete Applied Mathematics

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The Maximum Weight Independent Set (MWIS) problem on finite undirected graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum weight sum. MWIS is one of the most investigated and most important algorithmic graph problems; it is well known to be NP-complete, and it remains NP-complete even under various strong restrictions such as for trianglefree graphs. Its complexity was an open problem for P k -free graphs, k ≥ 5. Recently, Lokshtanov et al. [26] proved that MWIS can be solved in polynomial time for P 5 -free graphs, and Lokshtanov et al. [25] proved that MWIS can be solved in quasi-polynomial time for P 6 -free graphs. It still remains an open problem whether MWIS can be solved in polynomial time for P k -free graphs, k ≥ 6 or in quasi-polynomial time for P k -free graphs, k ≥ 7. Some characterizations of P k -free graphs and some progress are known in the literature but so far did not solve the problem. In this paper, we show that MWIS can be solved in polynomial time for (P 7 ,triangle)-free graphs. This extends the corresponding result for (P 6 ,triangle)-free graphs and may provide some progress in the study of MWIS for P 7 -free graphs.

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Section: Introductionsupporting

confidence: 80%

Maximum Weight Independent Sets for (P7,triangle)-free graphs in polynomial time

Brandstädt

Mosca

2018

Discrete Applied Mathematics

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“…More tractability results can be obtained if we put some additional restrictions on the instance graph, e.g., by forbidding more induced subgraphs [6,19,28,33,34,36]. A slightly different direction was considered by Lozin, Milanič, and Purcell [31], who proved that for every fixed t, MWIS is polynomial-time solvable in subcubic S t,t,1 -free graphs.…”

Section: Introductionmentioning

confidence: 99%

Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws

Abrishami¹,

Chudnovsky²,

Dibek³

et al. 2021

Preprint

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For graphs G and H, we say that G is H-free if it does not contain H as an induced subgraph. Already in the early 1980s Alekseev observed that if H is connected, then the Max Weight Independent Set problem (MWIS) remains NP-hard in H-free graphs, unless H is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory.A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in H-free graphs, where H is any fixed path. If H is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019].In this paper we make an important step towards solving the problem by showing that for any subdivided claw H, MWIS is polynomial-time solvable in H-free graphs of bounded degree.

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“…The class of Triangle-free has been considered in the context of similar previous manuscripts on other subclasses of Triangle-free graphs, namely, on (P 7 ,Triangle)free graphs [7] − see [27] for an extension of this result − and more generally on (S 1,2,4 ,Triangle)-free graphs [9]. However let us observe that Lozin's conjecture is open also for those S i,j,k -graphs for any fixed indices i, j, k which in addition are Triangle-free − recalling that WIS remains NPhard for Triangle-free graphs − that is for restricted and more studied graph classes.…”

Section: Introductionmentioning

confidence: 99%

Independent sets in ($P_4+P_4$,Triangle)-free graphs

Mosca¹

2020

Preprint

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The Maximum Weight Independent Set Problem (WIS) is a well-known NPhard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H, G + H denotes the disjoint union of G and H.This manuscript shows that (i) WIS can be solved for (P 4 + P 4 , Triangle)-free graphs in polynomial time, where a P 4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every (P 4 + P 4 , Triangle)-free graph G there is a family S of subsets of V (G) inducing (complete) bipartite subgraphs of G, which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of S. These results seem to be harmonic with respect to other polynomial results for WIS on certain [subclasses of] S i,j,k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.

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Maximum weight stable set in (P7, bull)-free graphs and (S1,2

Cited by 6 publications

References 27 publications

Maximum Weight Independent Sets for (P7,triangle)-free graphs in polynomial time

Maximum Weight Independent Sets for (P7,triangle)-free graphs in polynomial time

Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws

Independent sets in ($P_4+P_4$,Triangle)-free graphs

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