arXiv:2610.36502v1 Announce Type: new Abstract: Statistical graphical models are representations of the dependences between the components of a random vector. They encode these dependences via graphical separation criteria. We consider the algorithmic problem of checking separation statements for the various graph classes and separation criteria that have been developed by the community. We establish that all criteria we study can be checked by modular variations of reachability. For all separation criteria but $\sigma$-separation, this yields a linear-time algorithm. This is effectively the optimal attainable time since loading the graph itself is linear-time in the number of edges in the graph. Using the CIfly framework for deploying modular variations of reachability, we also provide R and Python code implementing our algorithms. We evaluate empirical runtime and scaling behavior in a simulation study.