Distance-locally disconnected graphs

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(n) 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 Research supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme. Research supported by grant No. P202/12/G061 of the Czech Science Foundation and by the European Regional Development Fund (ERDF), project NTIS New Technologies for Information Society, European Centre of Excellence, CZ.1.05/1.1.00/02.0090. 204 M. Miller, J. Ryan and Z. Ryjáček general upper bound and we show its asymptotic sharpness for some values of k. We also discuss some connections with cages.