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Discussiones Mathematicae Graph Theory 24(2) (2004) 319-343
DOI: https://doi.org/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,…, P_n be additive hereditary graph properties. A graph G has property (P₁º …º P_n) if there is a partition (V₁,…,V_n) 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 = 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

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Received 6 November 2002
Revised 24 January 2003