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Making a tournament $$k$$-arc-strong by reversing or deorienting arcs. (English) Zbl 1035.05045
Summary: We prove that every tournament $$T=(V,A)$$ on $$n \geq 2k+1$$ vertices can be made $$k$$-arc-strong by reversing no more than $$k(k+1)/2$$ arcs. This is best possible as the transitive tournament needs this many arcs to be reversed. We show that the number of arcs we need to reverse in order to make a tournament $$k$$-arc-strong is closely related to the number of arcs we need to reverse just to achieve in- and out-degree at least $$k$$. We also consider, for general digraphs, the operation of deorienting an arc which is not part of a 2-cycle. That is we replace an arc $$xy$$ such that $$yx$$ is not an arc by the 2-cycle $$xyx$$. We prove that for every tournament $$T$$ on at least $$2k+1$$ vertices, the number of arcs we need to reverse in order to obtain a $$k$$-arc-strong tournament from $$T$$ is equal to the number of arcs one needs to deorient in order to obtain a $$k$$-arc-strong digraph from $$T$$. Finally, we discuss the relations of our results to related problems and conjectures.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C40 Connectivity
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##### References:
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