Consistent input/output partitions in a causal systems

Many physical systems do not come with a natural input/output partition of the system variables: we call them a causal systems. There are advantages in studying a causal systems using approaches that do not need a partition of the system variables into inputs and outputs. It is also well-acknowledged that viewing subsystems as input/output blocks helps in our understanding of such systems. In view of this, it is helpful to count the number of input/output partitions that one can assign in a `consistent' way such that a causal systems can be analyzed and simulated using input/output approaches/tools. We introduce a notion of `consistent' input/output partitioning and provide a count for various networks made up of simple 2-terminal electrical blocks (resistors, inductors and capacitors) with one source. We relate the count for the ladder networks to the Fibonacci series. When the 2-terminal building blocks impose their own natural input/output partition, depending on the properness of the transfer function, then this count reduces suitably. This paper formulates and answers such enumeration questions for various common electrical networks. We use techniques from graph theory: of the type used in KCL/KVL/electrical-networks and matching theory in bipartite graphs. Using these techniques we convert a circuit topology into signal flow graphs (SFG) familiar in systems and control theory.