On the factorization of reducible properties of graphs into irreducible factors

Abstract

A hereditary property R of graphs is said to be reducible if there exist hereditary properties P 1 , P 2 such that G ∈ R if and only if the set of vertices of G can be partitioned into V (G) = V 1 ∪ V 2 so that V 1 ∈ P 1 and V 2 ∈ P 2. The problem of the factorization of reducible properties into irreducible factors is investigated.

Unique factorization theorem

“…have been stated in the book [8] of Jensen and Toft "Graph Coloring Problems". Partial results for some subclasses of inducedhereditary properties may be found in [11,12,9,13]. In May 1995 (see [11]) we proved the unique factorization theorem (UFT) for the additive hereditary properties with completeness at most 3, in June 1996 (see [9]) we proved UFT.…”

“…Partial results for some subclasses of inducedhereditary properties may be found in [11,12,9,13]. In May 1995 (see [11]) we proved the unique factorization theorem (UFT) for the additive hereditary properties with completeness at most 3, in June 1996 (see [9]) we proved UFT. The aim of this paper is to prove the unique factorization in the whole class of additive induced-hereditary properties of graphs.…”

A survey of hereditary properties of graphs

“…Thus for the class of outerplanar graphs we may have more reducible bounds. In [71] it has been proved that Since the structure of reducible properties of completeness c(R) ≥ 3 is very complicated (see [74]) there are only some partial results on rather simple properties with completeness 3 (see [11,66]). The proofs of these results are based on the following lemma.…”

“…It was proved that this factorization is unique also in L a : By Theorem 54 the answer is affirmative for hom-properties. In [74] the unique factorization of all additive and hereditary properties of completeness at most three has been proved. It turns out that Theorem 35 provides an important contribution to the solution of the mentioned general problem.…”