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- W2032599964 abstract "Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D ≤ 2l- 1, then G has maximum connectivity (κ = δ), and if D ≤ 2l, then it attains maximum edge-connectivity (λ = δ), where l is a parameter which can be thought of as a generalization of the girth of a graph. In this paper, we study some similar conditions for a digraph to attain high connectivities, which are given in terms of what we call the conditional diameter or P-diameter of G. This parameter measures how far apart can be a pair of subdigraphs satisfying a given property P, and, hence, it generalizes the standard concept of diameter. As a corollary, some new sufficient conditions to attain maximum connectivity or edge-connectivity are derived. It is also shown that these conditions can be slightly relaxed when the digraphs are bipartite. The case of (undirected) graphs is managed as a corollary of the above results for digraphs. In particular, since l ≥ 1, some known results of Plesnik and Znam are either reobtained or improved. For instance, it is shown that any graph whose line graph has diameter D = 2 (respectively, D ≤ 3) has maximum connectivity (respectively, edge-connectivity.) Moreover, for graphs with even girth and minimum degree large enough, we obtain a lower bound on their connectivities." @default.
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- W2032599964 date "1999-10-01" @default.
- W2032599964 modified "2023-10-15" @default.
- W2032599964 title "On the superconnectivity and the conditional diameter of graphs and digraphs" @default.
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- W2032599964 doi "https://doi.org/10.1002/(sici)1097-0037(199910)34:3<197::aid-net4>3.0.co;2-9" @default.
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