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

Brandstädt, Andreas; Mosca, Raffaele

doi:10.1016/j.dam.2017.10.003

Cited by 15 publications

(16 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then one can derive the following result, which is similar to the corresponding results obtained for (P 7 ,Triangle)-free graphs [7] and for (S 1;2;4 ,Triangle)-free graphs [9], and which seems to be harmonic [together with such results] with respect to the result of Prömel et al [39] showing that with ''high probability'' removing a single vertex in a Triangle-free graph leads to a bipartite graph.…”

Section: Proof Of Factsupporting

confidence: 79%

“…The class of Triangle-free graphs, which seems to be a more studied graph class (see e.g. [23]), 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], more generally on (P 7 ,Bull)-free and (S 1;2;3 ,Bull)-free graphs [29], and on (S 1;2;4 ,Triangle)-free graphs [9].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

Mosca

2021

Graphs and Combinatorics

Self Cite

View full text Add to dashboard Cite

The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard 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, $$G+H$$ G + H denotes the disjoint union of G and H. This manuscript shows that (i) WIS can be solved for ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graphs in polynomial time, where a $$P_4$$ 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 ($$P_4+P_4$$ P 4 + P 4 , Triangle)-free graph G there is a family $${{\mathcal {S}}}$$ 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 $${\mathcal {S}}$$ S . These results seem to be harmonic with respect to other polynomial results for WIS on [subclasses of] certain $$S_{i,j,k}$$ S i , j , k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.

show abstract

Section: Proof Of Factsupporting

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

Mosca

2021

Graphs and Combinatorics

Self Cite

View full text Add to dashboard Cite

show abstract

“…where the maximum is taken over all winning coalitions in W and all losing coalitions in L. A simple game (N, v) is a weighted voting game if and only if α < 1. 5 This follows from observing that each optimal solution p of (1) can be scaled to satisfy p(W ) ≥ 1 for all winning coalitions W .…”

Section: Introductionmentioning

confidence: 99%

Simple Games Versus Weighted Voting Games

Hof

Kern

Kurz

et al. 2018

Algorithmic Game Theory

View full text Add to dashboard Cite

A simple game (N, v) is given by a set N of n players and a partition of 2 N into a set L of losing coalitions L with value v(L) = 0 that is closed under taking subsets and a set W of winning coalitions W with v(W ) = 1. Simple games with α = min p≥0 maxW ∈W,L∈L p(L) p(W ) < 1 are known as weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that α ≤ 1 4 n for every simple game (N, v). We confirm this conjecture for two complementary cases, namely when all minimal winning coalitions have size 3 and when no minimal winning coalition has size 3. As a general bound we prove that α ≤ 2 7 n for every simple game (N, v). For complete simple games, Freixas and Kurz conjectured that α = O( √ n). We prove this conjecture up to a ln n factor. We also prove that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, computing α is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if α < a is polynomial-time solvable for every fixed a > 0.

show abstract

“…It is solvable in polynomial time for H -free graphs whenever H is contained in P k for k = 6 (see [20] for H P = 5 and [18] for H P = 6 ) or contained in S i j k , , with i j k ( , , ) (1, 1, 2) ≤ (see [5,22] for the weighted version). It is solvable in polynomial time for (P 7 , triangle)-free graphs [8] and for (S 1,2,4 , triangle)-free graphs [9]. The complexity is not known for H-free graphs whenever H is some S i j k , , that contains either P S , 7 1,1,3 , or S 1,2,2 .…”

Section: Algorithmic Consequencesmentioning

confidence: 99%

(Theta, triangle)‐free and (even hole, K4)‐free graphs. Part 2: Bounds on treewidth

Pilipczuk

Sintiari

Thomassé

et al. 2021

Journal of Graph Theory

View full text Add to dashboard Cite

A theta is a graph made of three internally vertexdisjoint chordless paths P a b P a b = … , = … 1 2 S i j k , , be the tree with a vertex v, from which start three paths with i j , , and k edges, respectively. We denote by K t the complete graph on t vertices. We prove that for all nonnegative integers i j k , , , the class of graphs that contain no theta, no K 3 , and no S i j k , , as induced subgraphs have bounded treewidth. We prove that for all nonnegative integers i j k t , , , , the class of graphs that contain no even hole, no pyramid, no K t , and no S i j k , , as induced subgraphs have bounded treewidth. To bound the treewidth, we prove that every graph of large treewidth must contain a large clique or a minimal separator of large cardinality.

show abstract

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

Cited by 15 publications

References 51 publications

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

Independent Sets in ($$P_4+P_4$$,Triangle)-Free Graphs

Simple Games Versus Weighted Voting Games

(Theta, triangle)‐free and (even hole, K4)‐free graphs. Part 2: Bounds on treewidth

Contact Info

Product

Resources

About