Problem Controlling a 4.5v DC motor with Arduino Uno

Here's a little intuition you would benefit from perhaps: A circuit is called a circuit because current goes round it.

If two circuits are connected by one wire rather than two, there isn't a circuit, no current is going to flow - they are not able to communicate. Its taken implicitly everywhere in electronics that there is a ground return for all signals.

Thus the OneWire bus uses two wires, the TWI (two wire interface) uses 3 wires really...

(Actually this is a slight over-simplification, since connecting two circuits with only one wire might involve a flow of current, but only for an instant as they reach a common potential - I've also glossed over capacitive coupling)