Complete graph-theoretic characterization of fixed nodes

Establish necessary and sufficient graph-theoretic conditions that completely characterize which state nodes are fixed nodes in linear structured systems represented by directed graphs, extending beyond currently known necessary conditions so as to yield a full characterization of fixed nodes.

Background

The paper studies controllability of individual nodes in structured networks and adopts the notion of a fixed controllable subspace. Prior work (Commault et al., 2017) introduced fixed nodes and provided tests based on input addition, while Van der Woude et al. (2019) gave conditions for non-fixed nodes and necessary conditions for fixed nodes using supremal minimal separators.

The authors note that those prior results do not provide a complete, necessary-and-sufficient graph-theoretic characterization of fixed nodes in general graphs. While this paper contributes necessary and sufficient conditions specifically for DAGs, the broader problem of full characterization for general directed graphs remains open.