  The causal revolution has stimulated interest in understanding complex
relationships in various fields. Most of the existing methods aim to discover
causal relationships among all variables within a complex large-scale graph.
However, in practice, only a small subset of variables in the graph are
relevant to the outcomes of interest. Consequently, causal estimation with the
full causal graph -- particularly given limited data -- could lead to numerous
falsely discovered, spurious variables that exhibit high correlation with, but
exert no causal impact on, the target outcome. In this paper, we propose
learning a class of necessary and sufficient causal graphs (NSCG) that
exclusively comprises causally relevant variables for an outcome of interest,
which we term causal features. The key idea is to employ probabilities of
causation to systematically evaluate the importance of features in the causal
graph, allowing us to identify a subgraph relevant to the outcome of interest.
To learn NSCG from data, we develop a necessary and sufficient causal
structural learning (NSCSL) algorithm, by establishing theoretical properties
and relationships between probabilities of causation and natural causal effects
of features. Across empirical studies of simulated and real data, we
demonstrate that NSCSL outperforms existing algorithms and can reveal crucial
yeast genes for target heritable traits of interest.
