This question is inspired by the physics of circuity.
I was troubled by the assumptions of high school physics that we only need to know 2 laws. 1 is parallel circuit laws and 2 is series circuit laws, essentially both are derived by the 'drag' equation for stable circuits (Ohm's Law) and the special case of maxwell equations/the 'potential' equation, (Kirchoff's Laws).
The problem is, you can build on more than just series and parallel, in fact, even when Electrical Engineers design the grid for street lights for example, they apply certain good approximations to assume that circuits are in fact parallel when they are not for ease of calculation.
My task is to build a 'finite' set of 'patterns' that EVERY circuit can be built from. I hope this makes sense. At first I thought it would be possible using this list, but later realised that in fact bridges can connect with other bridges too. This became too confusing for my brain to work through as I could not see if you really needed more 'patterns' or if there is some topological equivalence to the current set which i could not spot! 
I hope you can make this problem easier, give some hint or shed some light on what the maths behind it is.
Certainly the current generalised program for solving a resistor battery (not sure about capacitor and inductor) circuit is to apply a large set of kirchoffs laws linear algebraic system and to solve it. Of course, intuitively, we solve by breaking the circuits into 1+2 components as this is less brute force. If the finite list of patterns is short, this generalised method of solving circuits could be less computationally intensive and give a better intuition behind the flow of electricity. My intuition works well in electric circuits comprised of 1), 2) and 4) but breaks down with circuits 3) because the general effect of 3) has not been derived yet and is not discussed in college.