About weak connectivity (Q1356780)

A tournament is an orientation of a complete graph. A local tournament is an oriented graph \(D=(V,A)\) (i.e. without 2-cycles) with the property that for every vertex \(x\in V\) the set of out-neighbours as well as the set of in-neighbours of \(x\) induce a tournament. The theme of this paper are those digraphs such that for every induced subgraph, one can define (disjoint) weakly connected components. When oriented and finite these are exactly the local tournaments. The structure of these graphs when infinite and locally infinite is given in this paper. Finally it is shown that the general case (where 2-cycles are allowed) can be reduced to the oriented case by contraction of 2-cycles.