DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory  17(1) (1997)   51-66
DOI: 10.7151/dmgt.1038

MAXIMAL GRAPHS WITH RESPECT TO HEREDITARY PROPERTIES

Izak Broere

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

Marietjie Frick

Department of Mathematics, Applied Mathematics and Astronomy
University of South Africa
P.O. Box 392, Pretoria, 0001 South Africa

Gabriel Semanišin

Department of Geometry and Algebra
P.J.Šafárik University
041 54 Košice, Slovak Republic

Abstract

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P1, …,Pn be properties of graphs. We say that a graph G has property P1 º…ºPn if the vertex set of G can be partitioned into n sets V1, …,Vn such that the subgraph of G induced by Vi has property Pi; i = 1,…, n. A hereditary property R is said to be reducible if there exist two hereditary properties P1 and P2 such that R=P1ºP2. If P is a hereditary property, then a graph G is called P- maximal if G has property P but G+e does not have property P for every e ∈ E([`G]). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.

Keywords: hereditary property of graphs, maximal graphs, vertex partition.

1991 Mathematics Subject Classification: 05C15, O5C75.

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