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Distance-Locally Disconnected Graphs

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Miller M., Ryan J., Ryjáček Z.

Discussiones Mathematicae Graph Theory

2013

tom 33

nr 1

203-215

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.

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1 Discussiones Mathematicae Graph Theory

1 Miller M.

1 Ryan J.

1 Ryjáček Z.

1 2013

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Distance-Locally Disconnected Graphs