On triangle-free projective graphs

The 2-projection property was introduced by Corominas for ordered sets; it was generalized to other structures and arities by Pouzet, Rosenberg and Stone and independently by Davey, McKenzie, Nation and Palfy. We investigate triangle-free projective graphs and give a characterization for graphs that satisfy a certain condition on their cycles. In those cases, projectivity is equivalent to quasiprojectivity. As a corollary, we obtain a characterization of projective orders of height 1.