I want to know if there have been studies regarding the minimum set of vertices intersecting all triangles in a graph.
This is a particular case of feedback vertex set, in which we want to intersect all cycles in a graph. There is a famous conjecture of Albertson and Berman (M. O. Albertson and D. M. Berman, A conjecture on planar graphs. Graph Theory and Related Topics, Academic Press 1979), which states that at most half of its vertices suffices to intersect all cycles in a graph.
What I asking also can be seen as a vertex variation of a problem that asks for a set of edges intersecting all triangles in a graph. For this, there is a famous conjecture of Tuza which relates this parameter to its natural dual.
I want to know if there are similar conjectures, when we want is a set of vertices intersecting all triangles.