In project scheduling under processing times uncertainty, the Anchor-Robust Project Scheduling Problem is to find a baseline schedule of bounded makespan and a max-weight subset of jobs whose starting times are guaranteed. The problem was proven NP-hard even for budgeted uncertainty. In the present work we design mixed-integer programming (MIP) formulations that are valid for a variety of uncertainty sets encompassing budgeted uncertainty. A new dominance among solutions is proposed, resulting into an MIP formulation. This dominance-based formulation is shown to be tighter than all previously known formulations. We further study the combinatorial structure of the problem. We show that the dominance-based formulation yields a polyhedral characterization in non-trivial cases of budgeted uncertainty. In numerical experiments we investigate how the formulation performs on instances around the polyhedral characterization cases, for both budgeted uncertainty sets and more elaborate uncertainty sets involving several budgets.