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Discussiones Mathematicae Graph Theory 19(2) (1999) 167-174
DOI: https://doi.org/10.7151/dmgt.1093

FACTORIZATIONS OF PROPERTIES OF GRAPHS

Izak Broere and Samuel John Teboho Moagi Department of Mathematics Faculty of Science, Rand Afrikaans University P.O. Box 524, Auckland Park, 2006 South Africa e-mail: ib@na.rau.ac.za

Peter Mihók Mathematical Institute, Slovak Academy of Sciences Gresákova 6, Košice, Slovak Republic e-mail: mihok@Košice.upjs.sk

Roman Vasky Department of Geometry and Algebra Faculty of Science, P.J. Safárik University Jesenná 5, 041 54 Košice, Slovak Republic e-mail: vasky@rsl.sk

Abstract

A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs P₁,P₂,...,P_n a vertex (P₁, P₂, …,P_n)-partition of a graph G is a partition {V₁,V₂,...,V_n} of V(G) such that for each i = 1,2,...,n the induced subgraph G[V_i] has property P_i. The class of all graphs having a vertex (P₁, P₂, …,P_n)-partition is denoted by P₁ºP₂º…ºP_n. A property ℜ is said to be reducible with respect to a lattice of properties of graphs L if there are n ≥ 2 properties P₁,P₂,...,P_n ∈ L such that ℜ = P₁ºP₂º…ºP_n; otherwise ℜ is irreducible in L. We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.

Keywords: factorization, property of graphs, irreducible property, reducible property, lattice of properties of graphs.

1991 Mathematics Subject Classification: 05C15, O5C75.

References

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Received 2 February 1999
Revised 8 September 1999