Geodesic
Distance-Locally Disconnected Graphs
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 203-215

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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.
Keywords: neighborhood, distance, locally disconnected, cage 
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Miller, Mirka; Ryan, Joe; Ryjáček, Zdeněk. Distance-Locally Disconnected Graphs. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 1, pp. 203-215. http://geodesic.mathdoc.fr/item/DMGT_2013_33_1_a16/