  This paper explores the conditions for determining fixed nodes in structured
networks, specifically focusing on directed acyclic graphs (DAGs). We introduce
several necessary and sufficient conditions for determining fixed nodes in
$p$-layered DAGs. This is accomplished by defining the problem of maximum
disjoint stems, based on the observation that all DAGs can be represented as
hierarchical structures with a unique label for each layer. For structured
networks, we discuss the importance of fixed nodes by considering their
controllability against the variations of network parameters. Moreover, we
present an efficient algorithm that simultaneously performs labeling and fixed
node search for $p$-layered DAGs with an analysis of its time complexity. The
results presented in this paper have implications for the analysis of
controllability at the individual node level in structured networks.
