The critical behaviour of statistical models with long-range interactions exhibits distinct regimes as a function of ρ, the power of the interaction strength decay. For ρ large enough, ρ > ρ sr , the critical behaviour is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ρ sr in the case of the long-range XY model in two dimensions. We study the 2d XY model on the diluted graph, a sparse graph obtained from the 2d lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as |r| −ρ , which is expected to feature the same critical behavior of the long range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetisation and of the Binder cumulant, we present numerical evidence that ρ sr = 4. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless (BKT) universality class for ρ ≥ 4, and to present a 2 nd -order transition for ρ < 4. 1 arXiv:1905.06688v1 [cond-mat.stat-mech]