Extraconnectivity of graphs with large minimum degree and girth (original) (raw)

1997, Discrete Mathematics

The extraconnectivity κ(n) of a simple connected graph G is a kind of conditional connectivity which is the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has more than n vertices. The usual connectivity and superconnectivity of G correspond to κ(0) and κ(1) respectively. This paper gives sufficient conditions, relating the diameter D, the girth g, and the minimum degree δ of a graph, to assure maximum extraconnectivity. For instance, if D ≤ g − n + 2(δ − 3), being n ≥ 2δ + 4 and g ≥ n + 5, then the value of κ(n) is (n+1)δ−2n, which is optimal. The corresponding edge version of this result, to assure maximum edge-extraconnectivity λ(n), is also discussed.

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