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Spanning $$k$$-arc-strong subdigraphs with few arcs in $$k$$-arc-strong tournaments. (English) Zbl 1057.05039
It is shown that every $$k$$-arc-strong tournament $$T$$ contains a spanning $$k$$-arc-strong subdigraph with at most $$nk+136k^2$$ arcs. Moreover, this is best possible in terms of the exponent on $$k$$. Such a subdigraph can be found in polynomial time. The proof uses new results on digraphs in which the number of non-neighbors of each vertex is bounded by some constant $$c$$. It is conjectured that for every $$k$$-arc-strong tournament $$T$$ the minimum number of arcs in a $$k$$-arc-strong spanning subdigraph of $$T$$ is equal to the minimum number of arcs in a spanning subdigraph of $$T$$ with the property that every vertex has in- and outdegree at least $$k$$.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
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