Graph diffusion models have emerged as state-of-the-art techniques in graph generation, yet integrating domain knowledge into these models remains challenging. Domain knowledge is particularly important in real-world scenarios, where invalid generated graphs hinder deployment in practical applications.Unconstrained and conditioned graph generative models fail to guarantee such domain-specific structural properties. We present ConStruct, a novel framework that allows for hard-constraining graph diffusion models to incorporate specific properties, such as planarity or acyclicity.Our approach ensures that the sampled graphs remain within the domain of graphs that verify the specified property throughout the entire trajectory in both the forward and reverse processes. This is achieved by introducing a specific edge-absorbing noise model and a new projector operator.ConStruct demonstrates versatility across several structural and edge-deletion invariant constraints and achieves state-of-the-art performance for both synthetic benchmarks and attributed real-world datasets. For example, by leveraging planarity in digital pathology graph datasets, the proposed method outperforms existing baselines and enhances generated data validity by up to 71.1 percentage points.