  This paper presents new results and reinterpretation of existing conditions
for strong structural controllability in a structured network determined by the
zero/non-zero patterns of edges. For diffusively-coupled networks with
self-loops, we first establish a necessary and sufficient condition for strong
structural controllability, based on the concepts of dedicated and sharing
nodes. Subsequently, we define several conditions for strong structural
controllability across various graph types by decomposing them into disjoint
path graphs. We further extend our findings by introducing a composition rule,
facilitating the analysis of strong structural controllability in larger
networks. This rule allows us to determine the strong structural
controllability of connected graphs called pactus graphs (a generalization of
the well-known cactus graph) by consideration of the strong structural
controllability of its disjoint component graphs. In this process, we introduce
the notion of a component input node, which is a state node that functions
identically to an external input node. Based on this concept, we present an
algorithm with approximate polynomial complexity to determine the minimum
number of external input nodes required to maintain strong structural
controllability in a diffusively-coupled network with self-loops.
