Nearly-acyclically pushable tournaments

Let D be a digraph and X ~ V(D). By pushing X we mean reversing the orientation of each arc of D with exactly one end in X. Klostermeyer proved that it is NP-complete to decide if a given digraph can be made acyclic using the push operation. By contrast, Huang, MacGillivray, and Wood showed that the problem of deciding if a given multipartite tournament can be made acyclic using the push operation is solvable in polynomial time. We define a digraph to be nearly-acyclic if it is obtained from an acyclic digraph by substituting a (directed) triangle or a single vertex for each vertex of the acyclic digraph. It is shown that it is NPcomplete to decide if a given digraph can be made nearly-acyclic using the push operation. In this paper, we characterize, in terms of forbidden subtournaments, the tournaments which can be made nearly-acyclic by pushing. This implies that the problem of deciding if a given tournament can be made nearly-acyclic using the push operation is solvable in polynomial time.