Unique factorization theorem

Abstract

A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let P 1 , P 2 , . . . , P n be properties of graphs. A graph G isA property R is said to be reducible if there exist properties P 1 and P 2 such that R = P 1 •P 2 ; otherwise the property R is irreducible. We prove that every additive and inducedhereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (P 1 , P 2 , . . . , P n )-partitionable graphs for any irreducible properties P 1 , P 2 , . . . , P n .

Factorizations and characterizations of induced‐hereditary and compositive properties

“…We use this characterization here to give a new proof of the result of Mihók et al [19] that hereditary compositive and additive hereditary properties have a (unique) factorization into irreducible properties with the same restrictions. There are similar, but more difficult, results for indiscompositive and additive induced-hereditary properties [10,18]. We are grateful to Jim Geelen for suggesting this approach.…”

“…(Since it is rather more complicated, we do not discuss here the analogous concept for indiscompositive properties. However, it is in [11,18], and the indiscompositive factorization is given in the manner of this paper in [10].) A graph G is decomposable if it is the join of two graphs; otherwise, G is indecomposable.…”

“…The set L a of additive induced-hereditary properties has a unique factorization theorem [11,18] (see [2] for a generalization to hypergraphs), but there are properties in the set L of induced-hereditary properties that are not uniquely factorizable over L [19,Section 4]. Szigeti and Tuza [24,Prob.…”

“…The proofs in [11,18] are rather difficult, particularly for the technical point, which is at the heart of everything.…”

“…We give a Scheinerman-like characterization of additive induced-hereditary and indiscompositive properties in Section 3. The unique factorization results and proofs of Section 4 can also be generalized to these classes of properties, using the previously mentioned technical results in [18].…”