Abstract :
The η-extraconnectivity κη of a simple connected (di)graph G is a kind of conditional connectivity which is defined as the minimum cardinality of a set of vertices, whose deletion disconnects G, in such a way that all remaining (strongly) connected components have cardinality greater than η. The usual connectivity and superconnectivity of G correspond to κ0 and κ1, respectively. First, this paper gives sufficient conditions, relating the diameter D, the parameter s and the minimum degree δ of a s-geodetic digraph, to assure maximum η-extraconnectivity. To be more precise, it is proved that if D ⩽ 2s − 1 − 2⌊logδ(η(δ − 1) + 1)⌋, being η ⩾ 2 and s ⩾ η + 1, then the value of κη is (η + 1)(δ − 1), which is optimal. Finally, in the undirected case, it is proved, for instance, that if D ⩽ g − 6 − 4 log(δ − 1) η (δ − 2) + 2δ, being g ⩾ η + 5 the girth of graph, η ⩾ δ + 1, and δ ⩾ 5, then the value of κη is (η + 1)δ − 2η, which is optimal. So, knowing the sufficient conditions relating the diameter D, the girth g and the minimum degree δ of a graph, to assure maximum η-extraconnectivity are improved for η ⩾ 2(δ + 1). The corresponding edge version of these results, to assure maximum edge η-extraconnectivity λη, are also discussed.
