Here are two adjacency matrices of 13-node tournaments,
for which it is necessaary to reverse 28 edges to make them 
become acyclic:

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13x13 MATRIX #1  
(lower triangle shown; complete upper triangle by means of antisymmetry):

  0
 -1  0
  1 -1  0
 -1  1  1  0
  1  1 -1 -1  0
  1 -1  1  1 -1  0
  1 -1 -1  1 -1 -1  0
 -1 -1 -1 -1  1  1  1  0
 -1 -1 -1  1  1 -1  1  1  0
 -1 -1  1  1 -1  1 -1 -1  1  0
 -1  1 -1 -1  1  1 -1 -1  1 -1  0
  1  1  1 -1 -1 -1 -1  1 -1  1 -1  0
  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1  0

Best vertex order 6,11,9,12,2,0,3,7,10,4,1,58

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13x13 MATRIX #2:
  0
 -1  0
  1 -1  0
 -1  1  1  0
  1  1 -1 -1  0
  1 -1  1  1 -1  0
 -1 -1 -1  1 -1  1  0
 -1 -1 -1  1  1 -1 -1  0
 -1 -1 -1 -1  1  1  1 -1  0
  1 -1  1 -1  1 -1 -1  1 -1  0
 -1  1  1 -1 -1  1 -1 -1  1 -1  0
  1  1 -1  1 -1 -1  1 -1 -1  1 -1  0
  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1  0

best vertex order: 3,1,9,2,4,0,6,7,10,8,5,11,12

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Such matrices are extremely rare.  My computer examined over half of all
13x13 regular tournament matrices (of which there are 1495297 isomorphism classes) 
and found only these two examples.  The complete examination of all 
1495297 should require about 1 week of compute time.

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This is an extension of one of them to 14 vertices which
requires reversing 30 arcs to make it acyclic.

14
  0
 -1  0
  1 -1  0
 -1  1  1  0
  1  1 -1 -1  0
  1 -1  1  1 -1  0
 -1 -1 -1  1 -1  1  0
 -1 -1 -1  1  1 -1 -1  0
 -1 -1 -1 -1  1  1  1 -1  0
  1 -1  1 -1  1 -1 -1  1 -1  0
 -1  1  1 -1 -1  1 -1 -1  1 -1  0
  1  1 -1  1 -1 -1  1 -1 -1  1 -1  0
  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1  0
  1  1  1 -1 -1 -1  1  1  1 -1 -1 -1  1  0

best vertex order:  3,4,10,11,13,1,8,6,5,9,12,2,0,7

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-Warren D. Smith September 2006.