Abstract Developing accurate algorithms for learning structures of probabilistic graphical models is an important problem within modern AI research. Here we focus on score-based structure learning for Bayesian networks as arguably the most central class of graphical models. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on integer programming (IP), is implemented in the GOBNILP system. Despite the recent algorithmic advances, current understanding of foundational aspects underlying the IP based approach to BNSL is still somewhat lacking. In this paper, we provide theoretical contributions towards understanding fundamental aspects of cutting planes and the related separation problem in this context, ranging from NP-hardness results to analysis of polytopes and the related facets in connection to BNSL