Causal discovery seeks to unveil causal relationships (represented as a so-called causal graph) from observational data. This paper investigates the complex relationship between the graph structure and the efficiency of constraint-based causal discovery algorithms. Our main contributions include (i) a near-tight characterization of which causal graphs admit a small d-separating set for each pair of vertices and thus can potentially be efficiently recovered by a constraint-based causal discovery algorithm, (ii) the explicit construction of a sequence of causal graphs on which the influential PC algorithm might need exponential time, although there is a small d-separating set between every pair of variables, and (iii) the formulation of a new causal discovery algorithm which achieves fixed-parameter running time by considering the maximum number of edge-disjoint paths between variables in the (undirected) super-structure as the parameter. A distinguishing feature of our investigation is that it is carried out within a more fine-grained model which more faithfully captures the infeasibility of performing accurate independence tests for large sets of conditioning variables.