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It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem.

We give a polynomial delay algorithm for enumerating the minimal transversals of hypergraphs without induced cycles of length 3 and 4. As a corollary, we can enumerate, with polynomial delay, the vertices of any polyhedron P(A,1)={x in R^n | Ax >= 1, x >= 0}, when A is a balanced matrix that does not contain as a submatrix the incidence matrix of a cycle of length 4. Other consequences are a polynomial delay algorithm for enumerating the minimal dominating sets of graphs of girth at least 9 and an incremental delay algorithm for enumerating all the minimal dominating sets of a bipartite graph without induced 6 and 8-cycles.