Topological criteria for nonlinear resistive circuits containing controlled sources to have a unique solution

This paper gives a definitive solution to the following fundamental problem: When does a network containing nonlinear monotone resistors (characterized by strictly increasing onto function), dc sources (voltage and current sources), and linear controlled sources (all 4 types) possess a unique solution? Our uniqueness criteria is couched in strictly topological terms. In particular, the uniqueness of a large class of practical nonlinear circuits can be determined, often by inspection, by checking for the presence of a new and fundamental topological structure called a cactus graph.