Meet- and join-irreducibility of additive hereditary properties of graphs

Berger, Amelie J.; Broere, Izak; Moagi, Samuel John Teboho; Mihók, Peter

doi:10.1016/s0012-365x(01)00323-5

Cited by 7 publications

(7 citation statements)

References 5 publications

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“…The assertion of the next lemma was stated earlier in [8] for properties, which are {K 2 }-irreducible over L a ; the proof presented here imitates their proof.…”

Section: Uniquenesssupporting

confidence: 70%

“…The problems of {H}-reducibility, over L c and over other classes of graph properties were earlier discussed only for H = K 2 and H = K 2 [7,8,9,10,11,12,13]. The second one (see [8]) only over the class L a of additive hereditary graph properties (which is closed under disjoint unions of graphs) because in such a class {K 2 }-reducibility is equivalent to being a join of incomparable properties from the lattice (L a , ⊆) [1].…”

Section: C-reducibilitymentioning

confidence: 99%

“…The second one (see [8]) only over the class L a of additive hereditary graph properties (which is closed under disjoint unions of graphs) because in such a class {K 2 }-reducibility is equivalent to being a join of incomparable properties from the lattice (L a , ⊆) [1]. Properties which are {K 2 }-reducible over a given class of properties always have an irreducible {K 2 }-factorization over those class of graph properties unlike {H}-reducibility in general.…”

Section: C-reducibilitymentioning

confidence: 99%

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Unique factorization of compositive hereditary graph properties

Broere

Drgas-Burchardt

2011

Acta. Math. Sin.-English Ser.

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Abstract. A graph property is any class of graphs that is closed under isomorphisms. A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs G 1 , G 2 ∈ P there exists a graph G ∈ P containing both G 1 and G 2 as subgraphs.Let H be any given graph on vertices v 1 , . . . , v n , n ≥ 2. A graph property P is Hfactorizable over the class of graph properties P if there exist P 1 , . . . , P n ∈ P such that P consists of all graphs whose vertex sets can be partitioned into n parts, possibly empty, satisfying:(1) for each i the graph induced by the i th non-empty partition part is in P i , and (2) for each i and j with i = j there are no edges between the i th and j th parts if v i and v j are non-adjacent vertices in H. If a graph property P is H-factorizable over P and we know the graph properties P 1 , . . . , P n , then we write P = H[P 1 , . . . , P n ]. In such a case, the presentation H[P 1 , . . . , P n ] is called a factorization of P over P. This concept generalizes graph homomorphisms and (P 1 , . . . , P n )-colorings.In this paper we investigate all H-factorizations of a graph property P over the class of all hereditary compositive graph properties for finite graphs H. It is shown that in many cases there is exactly one such factorization.

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“…The assertion of the next lemma was stated earlier in [8] for properties, which are {K 2 }-irreducible over L a ; the proof presented here imitates their proof.…”

Section: Uniquenesssupporting

confidence: 70%

Section: C-reducibilitymentioning

confidence: 99%

Section: C-reducibilitymentioning

confidence: 99%

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Unique factorization of compositive hereditary graph properties

Broere

Drgas-Burchardt

2011

Acta. Math. Sin.-English Ser.

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“…The meet-and join-irreducible elements in the lattice M a have been characterized in [1]. There the authors prove the following two results: 2.…”

Section: Meet-and Join-irreducibilitymentioning

confidence: 89%

Prime ideals in the lattice of additive induced-hereditary graph properties

Berger¹,

Mihók

2003

Discuss. Math. Graph Theory

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An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded joinirreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type.

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“…For |J | = 2 Theorem 4 implies the assertion. For |J | ≥ 3, at least one of the properties H 1 [P 1 1 , . .…”

Section: Corollary 1 Letmentioning

confidence: 99%

On uniqueness of a general factorization of graph properties

Drgas-Burchardt¹

2009

Journal of Graph Theory

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Abstract:A graph property is any class of simple graphs, which is closed under isomorphisms. Let H be a given graph on vertices v 1 , . . . , v n . For graph properties P 1 , . . . , P n , we denote by H[P 1 , . . . , P n ] the class of those (P 1 , . . . , P n )-partitionable graphs G, with a corresponding vertex partition (V 1 , . . . , V n ), for which an edge {x i , x j } with x i ∈ V i and x j ∈ V j implies the existence of the edge {v i , v j } in the graph H. The problem of the unique description of a graph property P in the form H[P 1 , . . . , P n ] is investigated for P, P 1 , . . . , P n being from the class L a of all graph properties closed under taking disjoint unions and subgraphs. The unique factorization theorems obtained in the paper generalize known results of this type bringing together •-reducibility over L a and ∨-reducibility in the lattice (L a , ⊆). There is also offered a new insight into the modular decomposition tree for a graph.

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Meet- and join-irreducibility of additive hereditary properties of graphs

Cited by 7 publications

References 5 publications

Unique factorization of compositive hereditary graph properties

Unique factorization of compositive hereditary graph properties

Prime ideals in the lattice of additive induced-hereditary graph properties

On uniqueness of a general factorization of graph properties

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