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Discussiones Mathematicae Graph Theory 21(1) (2001) 111-117
DOI: https://doi.org/10.7151/dmgt.1136

MINIMAL FORBIDDEN SUBGRAPHS OF REDUCIBLE GRAPH PROPERTIES

Amelie J. Berger

Department of Mathematics
Rand Afrikaans University
P.O. Box 524, Auckland Park, 2006 South Africa
e-mail: abe@raua.rau.ac.za

Abstract

A property of graphs is any class of graphs closed under isomorphism. Let P₁, P₂, …,P_n be properties of graphs. A graph G is (P₁, P₂, …,P_n)-partitionable if the vertex set V(G) can be partitioned into n sets, { V₁, V₂, …, V_n}, such that for each i = 1,2,...,n, the graph G[V_i] ∈ P_i. We write P₁ ºP₂ º… ºP_n for the property of all graphs which have a (P₁, P₂, …,P_n)-partition. An additive induced-hereditary property ℜ is called reducible if there exist additive induced-hereditary properties P₁ and P₂ such that ℜ = P₁ºP₂. Otherwise ℜ is called irreducible. An additive induced-hereditary property P can be defined by its minimal forbidden induced subgraphs: those graphs which are not in P but which satisfy that every proper induced subgraph is in P. We show that every reducible additive induced-hereditary property has infinitely many minimal forbidden induced subgraphs. This result is also seen to be true for reducible additive hereditary properties.

Keywords: reducible graph properties, forbidden subgraphs, induced subgraphs.

2000 Mathematics Subject Classification: 05C55, 05C15.

References

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Received 7 October 2000