Graphs & digraphs (2nd ed.)
Dense graphs with cycle neighborhoods
Journal of Combinatorial Theory Series B
On graphs with a local hereditary property
Discrete Mathematics
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Let P be a family of graphs. A graph G is said to satisfy a property P locally if G[N(υ)] ∈ P for every υ ∈ V(G). The class of graphs that satisfies 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(Jk)) for k ≥ 1. Also, we give sat(n,L(Ok)) and sat(n,L(Jk)) for k = 1,2.