Prime ideals in the lattice of additive induced-hereditary graph properties

• Autorzy: Amelie J. Berger , Peter Mihók
• Języki publikacji: EN

Abstrakty

EN

An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded join-irreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type.

Słowa kluczowe

EN

hereditary graph property; prime ideal; distributive lattice; induced subgraphs

Kategorie tematyczne

• 06B10: Ideals, congruence relations
• 06D10: Complete distributivity
• 05C99: None of the above, but in this section

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2003
• Tom: 23
• Numer: 1
• Strony: 117-127

Daty

wydano

2003

otrzymano

2001-07-12

poprawiono

2002-07-29

Twórcy

autor

Amelie J. Berger

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

autor

Peter Mihók

• Departement of Applied Mathematics and Informatics, Faculty of Economics, University of Technology, B. Nĕmcovej 32, 040 02 Košice, Slovakia
• Mathematical Institute of Slovak Academy of Sciences, Gresákova 6, 040 01 Košice, Slovakia

Bibliografia

• [1] A. Berger, I. Broere, P. Mihók and S. Moagi, Meet- and join-irreducibility of additive hereditary properties of graphs, Discrete Math. 251 (2002) 11-18, doi: 10.1016/S0012-365X(01)00323-5.
• [2] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, Survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.
• [3] M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, ed., Advances in Graph Theory (Vishwa International Publication, Gulbarga, 1991) 41-68.
• [4] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove and D.S. Scott, A Compendium of Continuous Lattices (Springer-Verlag, 1980).
• [5] G. Grätzer, General Lattice Theory (Second edition, Birkhäuser Verlag, Basel, Boston, Berlin 1998).
• [6] J. Jakubík, On the lattice of additive hereditary properties of finite graphs, Discuss. Math. General Algebra and Applications 22 (2002) 73-86.
• [7] T.R. Jensen and B. Toft, Graph Colouring Problems (Wiley-Interscience Publications, New York, 1995).
• [8] E.R. Scheinerman, Characterizing intersection classes of graphs, Discrete Math. 55 (1985) 185-193, doi: 10.1016/0012-365X(85)90047-0.
• [9] E.R. Scheinerman, On the structure of hereditary classes of graphs, J. Graph Theory 10 (1986) 545-551.

Identyfikatory

DOI

10.7151/dmgt.1189