Some extremal problems of graphs with local constraints

Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N (v)]∈P for every v∈V (G). The class of graphs that satis6es the property P locally will be denoted by L(P) and we shall call such a class a local property. Let P be a hereditary property. A graph is said to be maximal with respect to a hereditary property P (shortly P-maximal) if it belongs to P and none of its proper supergraphs of the same order has the property P. A graph is P-extremal if it has the maximum number of edges among all P-maximal graphs of given order. This number is denoted by ex(n;P). If the number of edges of a P-maximal graph of order n is minimum, then the graph is called P-saturated and its number of edges is denoted by sat(n;P). In this paper, we shall describe the numbers ex(n; L(Ok)) and ex(n; L(Sk)) for k¿ 1. Also, we give sat(n; L(Ok)) and sat(n; L(Sk)) for k = 1; 2. c © 2002 Elsevier Science B.V. All rights reserved.