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Let D be a strongly connected digraph of order n. The kth power D k of D is the digraph with the same set of vertices, a vertex x being joined to a vertex y≠x in D k if the directed distance from x to y in D is less than or equal to k. We show that for k⩾⌊n/2⌋,Dk is traceable and we characterize the digraphs D of even order n whose (n/2−1)th power is traceable.
The effects of the vertex push operation (when a vertex is pushed, the direction of each of its incident arcs is reversed) on the connectivity of tournaments are studied. It is shown that there exists an n k such that any tournament with n vertices, n⩾n k , can be made k-connected using pushes, for any k⩾1. These results imply, for example, that sufficiently large tournaments can be...
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