Minimal forbidden subgraphs of reducible graph properties

• Autorzy: Amelie J. Berger
• Języki publikacji: EN

Abstrakty

EN

A property of graphs is any class of graphs closed under isomorphism. Let 𝓟₁,𝓟₂,...,𝓟ₙ be properties of graphs. A graph G is (𝓟₁,𝓟₂,...,𝓟ₙ)-partitionable if the vertex set V(G) can be partitioned into n sets, {V₁,V₂,..., Vₙ}, such that for each i = 1,2,...,n, the graph $G[V_i] ∈ 𝓟_i$. We write 𝓟₁∘𝓟₂∘...∘𝓟ₙ for the property of all graphs which have a (𝓟₁,𝓟₂,...,𝓟ₙ)-partition. An additive induced-hereditary property 𝓡 is called reducible if there exist additive induced-hereditary properties 𝓟₁ and 𝓟₂ such that 𝓡 = 𝓟₁∘𝓟₂. 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.

Słowa kluczowe

EN

reducible graph properties; forbidden subgraphs; induced subgraphs

Kategorie tematyczne

• 05C15: Coloring of graphs and hypergraphs
• 05C55: Generalized Ramsey theory

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2001
• Tom: 21
• Numer: 1
• Strony: 111-117

Daty

wydano

2001

otrzymano

2000-10-07

Twórcy

autor

Amelie J. Berger

• Department of Mathematics, Rand Afrikaans University, P.O. Box 524, Auckland Park, 2006 South Africa

Bibliografia

• [1] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.
• [2] P. Erdős and A. Hajnal, On chromatic number of graphs and set systems, Acta Math. Acad. Sci. Hungar. 17 (1966) 61-99, doi: 10.1007/BF02020444.
• [3] J. Nesetril and V. Rödl, Partitions of vertices, Comment Math. Universitatis Carolinae 17 (1) (1976) 85-95.

Identyfikatory

DOI

10.7151/dmgt.1136