DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 24(2) (2004) 319-343
DOI: 10.7151/dmgt.1234

UNIQUE FACTORISATION OF ADDITIVE INDUCED-HEREDITARY PROPERTIES

Alastair Farrugia and R. Bruce Richter

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

e-mail: afarrugia@math.uwaterloo.ca
e-mail: brichter@math.uwaterloo.ca

Abstract

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 (P1º …º Pn) 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 Pi. A property P is reducible if there are properties Q, R such that P = Qº 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.

Keywords: additive and hereditary graph classes, unique factorization.

2000 Mathematics Subject Classification: 05C70.

References

[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.

Received 6 November 2002
Revised 24 January 2003