  We develop terminology and methods for working with maximally oriented
partially directed acyclic graphs (maximal PDAGs). Maximal PDAGs arise from
imposing restrictions on a Markov equivalence class of directed acyclic graphs,
or equivalently on its graphical representation as a completed partially
directed acyclic graph (CPDAG), for example when adding background knowledge
about certain edge orientations. Although maximal PDAGs often arise in
practice, causal methods have been mostly developed for CPDAGs. In this paper,
we extend such methodology to maximal PDAGs. In particular, we develop
methodology to read off possible ancestral relationships, we introduce a
graphical criterion for covariate adjustment to estimate total causal effects,
and we adapt the IDA and joint-IDA frameworks to estimate multi-sets of
possible causal effects. We also present a simulation study that illustrates
the gain in identifiability of total causal effects as the background knowledge
increases. All methods are implemented in the R package pcalg.
