Abstract
Given a tournament matrix T, its reversal indexiR (T), is the minimum k such that the reversal of the orientation of k arcs in the directed graph associated with T results in a reducible matrix. We give a formula for iR (T) in terms of the score vector of T which generalizes a simple criterion for a tournament matrix to be irreducible. We show that iR (T)≤[(n−1)/2] for any tournament matrix T of order n, with equality holding if and only if T is regular or almost regular, according as n is odd or even. We construct, for each k between 1 and [(n−1)/2], a tournament matrix of order n whose reversal index is k. Finally, we suggest a few problems.
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