Discussiones Mathematicae Graph Theory 17(1) (1997)
51-66
DOI: https://doi.org/10.7151/dmgt.1038
MAXIMAL GRAPHS WITH RESPECT TO HEREDITARY PROPERTIES
Izak Broere Department of Mathematics |
Marietjie Frick Department of Mathematics, Applied Mathematics and
Astronomy |
Gabriel Semanišin Department of Geometry and Algebra |
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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