Unique factorisation of additive induced-hereditary properties
Alastair Farrugia ; R. Bruce Richter
Discussiones Mathematicae Graph Theory, Tome 24 (2004), p. 319-343 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ₁,...,ₙ be additive hereditary graph properties. A graph G has property (₁∘...∘ₙ) if there is a partition (V₁,...,Vₙ) of V(G) into n sets such that, for all i, the induced subgraph G[Vi] is in i. A property is reducible if there are properties , such that = ∘ ; otherwise it is irreducible. Mihók, Semanišin and Vasky [8] gave a factorisation for any additive hereditary property into a given number dc() of irreducible additive hereditary factors. Mihók [7] 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.

Publié le : 2004-01-01
EUDML-ID : urn:eudml:doc:270750 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1234,
     author = {Alastair Farrugia and R. Bruce Richter},
     title = {Unique factorisation of additive induced-hereditary properties},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {24},
     year = {2004},
     pages = {319-343},
     zbl = {1061.05070},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1234}
}

Alastair Farrugia; R. Bruce Richter. Unique factorisation of additive induced-hereditary properties. Discussiones Mathematicae Graph Theory, Tome 24 (2004) pp. 319-343. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1234/

[000] [1] I. Broere and J. Bucko, Divisibility in additive hereditary properties and uniquely partitionable graphs, Tatra Mt. Math. Publ. 18 (1999) 79-87. | Zbl 0951.05034

[001] [2] I. Broere, M. Frick and G. Semanišin, Maximal graphs with respect to hereditary properties, Discuss. Math. Graph Theory 17 (1997) 51-66, doi: 10.7151/dmgt.1038. | Zbl 0902.05027

[002] [3] A. Farrugia, Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard, submitted. | Zbl 1053.05046

[003] [4] A. Farrugia and R.B. Richter, Complexity, uniquely partitionable graphs and unique factorisation, in preparation. www.math.uwaterloo.ca/∼afarrugia/

[004] [5] A. Farrugia and R.B. Richter, Unique factorisation of induced-hereditary disjoint compositive properties, Research Report CORR 2002-ZZ (2002) Department of Combinatorics and Optimization, University of Waterloo. www.math.uwaterloo.ca/~afarrugia/. | Zbl 1061.05070

[005] [6] J. Kratochvil and P. Mihók, Hom-properties are uniquely factorizable into irreducible factors, Discrete Math. 213 (2000) 189-194, doi: 10.1016/S0012-365X(99)00179-X. | Zbl 0949.05025

[006] [7] P. Mihók, Unique Factorization Theorem, Discuss. Math. Graph Theory 20 (2000) 143-153, doi: 10.7151/dmgt.1114. | Zbl 0968.05032

[007] [8] P. Mihók, G. Semanišin and R. Vasky, Additive and hereditary properties of graphs are uniquely factorizable into irreducible factors, J. Graph Theory 33 (2000) 44-53, doi: 10.1002/(SICI)1097-0118(200001)33:1<44::AID-JGT5>3.0.CO;2-O | Zbl 0942.05056

[008] [9] G. Semanišin, On generating sets of hereditary properties, unpublished manuscript.

[009] [10] J. Szigeti and Zs. Tuza, Generalized colorings and avoidable orientations, Discuss. Math. Graph Theory 17 (1997) 137-146, doi: 10.7151/dmgt.1047. | Zbl 0908.05039