On the X-join decomposition for undirected graphs

Habib, Michel; Maurer, Markus

doi:10.1016/0166-218x(79)90043-x

Cited by 39 publications

(22 citation statements)

References 11 publications

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“…The idea of modular decomposition has been first described in the 1960s by Gallai [22], and also appeared in the literature under various other names such as prime tree decomposition [18], X -join decomposition [23], or substitution decomposition [34]. This technique allows one to reduce many graph problems from arbitrary graphs to the so-called prime graphs.…”

Section: Modular Decompositionmentioning

confidence: 98%

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

Lozin

Milanič

2008

Journal of Discrete Algorithms

102

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The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e., finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs.

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Section: Modular Decompositionmentioning

confidence: 98%

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

Lozin

Milanič

2008

Journal of Discrete Algorithms

102

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“…This generalizes the concept of fragility studied by Habib and Maurer [11] in the context of module systems of graphs. In view of [5], if pV, Qq is an interval system, then any nonfragile interval A P Q is a strong set.…”

Section: Preliminariesmentioning

confidence: 71%

Interval Decomposition Lattices are Balanced

Foldes

Radeleczki

2016

Demonstratio Mathematica

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Abstract. Intervals in binary or n-ary relations or other discrete structures generalize the concept of an interval in a linearly ordered set. They are defined abstractly as closed sets of a closure system on a set, satisfying certain axioms. Join-irreducible partitions into intervals are characterized in the lattice of all interval decompositions. This result is used to show that the lattice of interval decompositions is balanced, and the case when this lattice is distributive is also characterised.

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“…There have been a number of O n 4 , O n 3 , O nm , and O n 2 algorithms for finding modular decomposition, such as Buer and Möhring [1], Ehrenfeucht et al [8], Golumbic [11], Habib and Maurer [14], McConnell [17], Muller and Spinrad [24], Steiner [26], some of them for special cases or generalizations of the problem. The cotree decomposition of cographs and the series-parallel decomposition of series-parallel partial orders are special cases on graphs and digraphs, respectively, for which linear-time solutions have been given (see Corneil et al [3]; Valdes et al [28]).…”

Section: Introductionmentioning

confidence: 99%