Polynomial-time safe-set algorithm for clique-acyclic digraphs

A clique acyclic digraph is a digraph containing no directed triangle. Its independence number is the maximum size of a set of pairwise nonadjacent vertices. For a fixed constant independence number $\alpha$, there exists a polynomial-time algorithm to find a minimum safe set in such a digraph.

Safe-set algorithm conjecture. There exists a polynomial-time algorithm to find a minimum safe set in a clique acyclic digraph with a constant independence number $\alpha$.

The conjecture is motivated by a bound of Gyárfás et al. on the domination number of clique-acyclic digraphs with bounded independence number, together with the argument used earlier in the paper. Whether minimum safe sets can be found in polynomial time under these restrictions remains open.