Additive and hereditary properties of graphs are uniquely factorizable into irreducible factors

Mih�k, Peter; Semani?in, Gabriel; Vasky, Roman

doi:10.1002/(sici)1097-0118(200001)33:1<44::aid-jgt5>3.0.co;2-o

Cited by 10 publications

(19 citation statements)

References 6 publications

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“…In [7], Mihók gave a remarkably general construction of uniquely partitionable graphs, and used this to produce a factorisation for the wider class of properties that are additive and induced-hereditary. This was claimed to be unique using the same argument as in [8]. We generalise his construction, and our own results (Theorems 4.9, 5.2 and 5.3, respectively) to prove that this factorisation is in fact unique.…”

Section: Introductionsupporting

confidence: 55%

“…We note that unique factorisation was settled completely in [6] for a significant class of additive hereditary properties, the proof depending on the structure of those properties (and in the spirit of the proof we give here). It is possible to use the structure of the factorisation presented in [8] to show that any factorisation with exactly dc(P) additive hereditary factors must be the one constructed in that article (a similar proof is possible for the factorisation of [7]); the appeal of the proofs of uniqueness given here is that they are independent of the structure of the factors of P. Thus, they depend only on the more elementary aspects of [8] and [7].…”

Section: Introductionmentioning

confidence: 84%

“…Mihók, Semanišin and Vasky [8] gave a factorisation of an additive hereditary property P into a given number dc(P) of irreducible additive hereditary factors. This factorisation was shown to be well-defined, but it was also claimed to be unique.…”

Section: Introductionmentioning

confidence: 99%

“…In the next section we reproduce the essential concepts, definitions and results adapted from [8]; stating those results in a stronger fashion here, and sometimes omitting their simple proofs. Our own techniques and proofs are presented in Section 3.…”

Section: Introductionmentioning

confidence: 99%

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Unique factorisation of additive induced-hereditary properties

Farrugia

Richter²

2004

Discuss. Math. Graph Theory

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An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let P 1 , . . . , P n be additive hereditary graph properties. A graph G has property (P 1 • · · · • P n ) if there is a partition (V 1 , . . . , V n ) of V (G) into n sets such that, for all i, the induced subgraphSemanišin and Vasky [J. Graph Theory 33 (2000), 44-53] gave a factorisation for any additive hereditary property P into a given number dc(P) of irreducible additive hereditary factors. Mihók [Discuss. Math. Graph Theory 20 (2000), 143-153] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and disjoint unions).Their results left open the possiblity of different factorisations, maybe even with a different number of factors; we prove here that the given factorisations are, in fact, unique.

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Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unique factorisation of additive induced-hereditary properties

Farrugia

Richter²

2004

Discuss. Math. Graph Theory

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“…This is just (D ∩ P)-recognition; if D and P are both additive induced-hereditary, then so is D ∩ P, with F (D ∩ P) = min ≤ (F (D) ∪ F (P)). We leave it as an open question, for reducible P, to determine when D ∩P is also reducible; Mihók's characterisations [19,20] of reducibility may prove useful.…”

Section: New Directionsmentioning

confidence: 99%

Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

Farrugia¹

2004

Electron. J. Combin.

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Can the vertices of a graph G be partitioned into A ∪ B, so that G[A] is a line-graph and G[B] is a forest? Can G be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are just special cases of our result: if P and Q are additive induced-hereditary graph properties, then (P, Q)-colouring is NP-hard, with the sole exception of graph 2-colouring (the case where both P and Q are the set O of finite edgeless graphs). Moreover, (P, Q)-colouring is NP-complete iff Pand Q-recognition are both in NP. This proves a conjecture of Kratochvíl and Schiermeyer.

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Factorizations and characterizations of induced‐hereditary and compositive properties

Farrugia¹,

Mihók²,

Richter³

et al. 2005

Journal of Graph Theory

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A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If P and Q are properties, the product P • Q consists of all graphs G for which there is a partition of the vertex set of G into (possibly empty) subsets A and B with G[A] ∈ P and G[B] ∈ Q. A property is * The results presented here are part of the first author's Ph.D. thesis, written under the supervision of the second author. Jim Geelen suggested one of the main results of this paper.† The first author's doctoral studies in Canada were fully funded by the Canadian government through a Canadian Commonwealth Scholarship.‡ Research supported by NSERC.§ Research supported in part by Slovak VEGA Grant 1/0424/03. reducible if it is the product of two other properties, and irreducible otherwise.We completely describe the few reducible induced-hereditary properties that have a unique factorisation into irreducibles. Analogs of compositive and additive induced-hereditary properties are introduced and characterised in the style of Scheinerman [Discrete Math. 55 (1985) 185-193]. One of these provides an alternative proof that an additive hereditary property factors into irreducible additive hereditary properties.

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Additive and hereditary properties of graphs are uniquely factorizable into irreducible factors

Cited by 10 publications

References 6 publications

Unique factorisation of additive induced-hereditary properties

Unique factorisation of additive induced-hereditary properties

Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties is NP-hard

Factorizations and characterizations of induced‐hereditary and compositive properties

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