The causal graph is a directed graph that describes the variable dependencies present in a planning instance. A number of papers have studied the causal graph in both practical and theoretical settings. In this work, we systematically study the complexity of planning restricted by the causal graph. In particular, any set of causal graphs gives rise to a subcase of the planning problem. We give a complete classification theorem on causal graphs, showing that a set of graphs is either polynomial-time tractable, or is not polynomial-time tractable unless an established complexity-theoretic assumption fails; our theorem describes which graph sets correspond to each of the two cases. We also give a classification theorem for the case of reversible planning, and discuss the general direction of structurally restricted planning.