A polynomial algorithm to find an independent set of maximum weight in a fork-free graph

Lozin, Vadim V.; Milanič, Martin

doi:10.1016/j.jda.2008.04.001

Cited by 102 publications

(90 citation statements)

References 38 publications

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“…Now assume g is adjacent to one of v 2 , v p , say to v p , and again denote the third neighbor of g by v j . If j ∈ {p − 2, p − 1}, then we obtain conguration (5) or (6). If j ∈ {2, 3}, then we obtain conguration (7) or (8) Lemma 3.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Whether this condition is sucient for polynomial-time solvability of the problem is a big open question. Without the restriction on vertex degree, polynomial-time solvability of the problem in classes of S i,j,k -free graphs was shown only for very small values of i, j, k. In particular, the problem can be solved for S 1,1,1 -free (claw-free) graphs [11], S 1,1,2 -free (fork-free) graphs [5], and S 0,1,1 +S 0,1,1 -free (2P 3 -free) graphs [7]. The complexity of the problem in S 0,2,2 -free (P 5 -free) graphs remains an open problem in spite of the multiple partial results on this topic (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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On the Maximum Independent Set Problem in Subclasses of Planar Graphs

Milanič¹

2010

JGAA

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Abstract. It is known that the maximum independent set problem is NP-complete for subcubic graphs, i.e. graphs of vertex degree at most 3. Moreover, the problem is NP-complete for H-free subcubic graphs whenever H contains a connected component which is not a tree with at most 3 leaves. We show that if every connected component of H is a tree with at most 3 leaves and at most 7 vertices, then the problem can be solved for H-free subcubic graphs in polynomial time.

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Section: Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Maximum Independent Set Problem in Subclasses of Planar Graphs

Milanič¹

2010

JGAA

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“…On the other hand, by Lemma 3, a maximum independent set of P 3 (G) corresponds to a CD solution for G. Since maximum independent set problem can be solved in polynomial time in claw-free graphs [12], CD can be solved in polynomial time in (kite, house, xbanner,diamond)-free graphs.…”

Section: Is a CD Solution For G If And Only If Ementioning

confidence: 99%

“…In the literature, there are several graph transformation, among them the widely used line graph [12]. A graph H is a line graph of a graph G if the vertices of H are in a one-to-one correspondence with the edges of G, with two vertices being adjacent in H if and only if the corresponding edges of G have a vertex in common.…”

Section: Introductionmentioning

confidence: 99%

A new polynomial class of cluster deletion problem

Malek

Naanaa

2016

Annals of Computer Science and Information Systems

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Abstract-Cluster Deletion (CD) problem asks to transform a given graph into a cluster graph by at most k edge deletions. CD is a combinatorial problem arising in the field of classification. In this paper, we introduce a graph transformation which enabled the identification of new polynomially solvable classes of CD problem. We show that if a graph is K3-free or (diamond, kite, house, xbanner)-free then cluster deletion problem can be solved in polynomial time on that graph.

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“…It was applied frequently for the solution of N P-hard problems on special graph classes, e.g. by Lozin and Milanič (2008) for the solution of the maximum weight independent set on fork-free graphs.…”

Section: Modular Decomposition and Clique Separatorsmentioning

confidence: 99%

Approximation of knapsack problems with conflict and forcing graphs

Pferschy

Schauer

2016

J Comb Optim

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We study the classical 0-1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is the use of conflict (resp. forcing) graphs. By modifying a recent result of Lokstanov et al. (Proceedings of the 25th annual ACM-SIAM symposium on discrete algorithms, SODA, pp 570-581, 2014) we derive a fairly complicated FPTAS for the knapsack problem on weakly chordal conflict graphs. Next, we show that the techniques of modular decompositions and clique separators, widely used in the literature for solving the independent set problem on special graph classes, can be applied to the knapsack problem with conflict graphs. In particular, we can show that every positive approximation result for the atoms of prime graphs arising from such a decomposition carries over to the original graph. We point out a number of structural results from the literature which can be used to show the existence of an FPTAS for several graph classes characterized by the exclusion of certain induced subgraphs. Finally, a PTAS for the knapsack problem with H-minor free conflict graph is derived. This includes planar graphs and, more general, graphs of bounded genus. forcing graphs can be transformed into a minimization knapsack problem with conflict graphs. It follows immediately that all our FPTAS results of the current and a previous paper carry over from conflict graphs to forcing graphs. In contrast, the forcing graph variant is already inapproximable on planar graphs.

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A polynomial algorithm to find an independent set of maximum weight in a fork-free graph

Cited by 102 publications

References 38 publications

On the Maximum Independent Set Problem in Subclasses of Planar Graphs

On the Maximum Independent Set Problem in Subclasses of Planar Graphs

A new polynomial class of cluster deletion problem

Approximation of knapsack problems with conflict and forcing graphs

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