Unique factorisation of additive induced-hereditary properties

• Autorzy: Alastair Farrugia , R. Bruce Richter
• Języki publikacji: EN

Abstrakty

EN

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let 𝓟₁,...,𝓟ₙ be additive hereditary graph properties. A graph G has property (𝓟₁∘...∘𝓟ₙ) if there is a partition (V₁,...,Vₙ) of V(G) into n sets such that, for all i, the induced subgraph $G[V_i]$ is in $𝓟_i$. A property 𝓟 is reducible if there are properties 𝓠, 𝓡 such that 𝓟 = 𝓠 ∘ 𝓡 ; otherwise it is irreducible. Mihók, Semanišin and Vasky [8] gave a factorisation for any additive hereditary property 𝓟 into a given number dc(𝓟) 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.

Słowa kluczowe

EN

additive and hereditary graph classes; unique factorization

Kategorie tematyczne

• 05C70: Factorization, matching, partitioning, covering and packing

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2004
• Tom: 24
• Numer: 2
• Strony: 319-343

Daty

wydano

2004

Twórcy

autor

Alastair Farrugia

• Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada, N2L 3G1

autor

R. Bruce Richter

• Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada, N2L 3G1

Bibliografia

• [1] I. Broere and J. Bucko, Divisibility in additive hereditary properties and uniquely partitionable graphs, Tatra Mt. Math. Publ. 18 (1999) 79-87.
• [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.
• [3] A. Farrugia, Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard, submitted.
• [4] A. Farrugia and R.B. Richter, Complexity, uniquely partitionable graphs and unique factorisation, in preparation. www.math.uwaterloo.ca/∼afarrugia/
• [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/.
• [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.
• [7] P. Mihók, Unique Factorization Theorem, Discuss. Math. Graph Theory 20 (2000) 143-153, doi: 10.7151/dmgt.1114.
• [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
• [9] G. Semanišin, On generating sets of hereditary properties, unpublished manuscript.
• [10] J. Szigeti and Zs. Tuza, Generalized colorings and avoidable orientations, Discuss. Math. Graph Theory 17 (1997) 137-146, doi: 10.7151/dmgt.1047.

Identyfikatory

DOI

10.7151/dmgt.1234