Distance-Locally Disconnected Graphs
Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 1, p. 203.

Voir la notice de l'article dans European Digital Mathematics Library

For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs, we show that this asymptotic rate of growth cannot be achieved. For regular graphs, we give a general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.
Classification : 05C12, 05C35, 05C40
Mots-clés : neighborhood, distance, locally disconnected, cage, neighborhood distance 
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Mirka Miller; Joe Ryan; Zdeněk Ryjáček. Distance-Locally Disconnected Graphs. Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 1, p. 203. https://geodesic-test.mathdoc.fr/item/DMGT_2013__33_1_268130/