How to distinguish independent capacitors in an n-order circuit?
In a circuit, any closed loop of capacitors only, will decrease by 1 the number of independent capacitors.
Infact, if you know the voltage Vi(t) of n-1 capacitors, the voltage Vn(t) of capacitor n is automatically determined by Kirchoff Voltage Law around that loop.
The number of poles decreases by 1 as well.

In a circuit, any closed loop of capacitors and voltage sources will decrease by 1 the number of independent capacitors.
Example
In a circuit where we have a series of 2 capacitors and a voltage source, the number of independent capacitors is 1.