Showing that its always possible to convert the circuit into the below equivalent form Conside the small circuit ( which is a part of a big circuit ) in which there are many connection of various resistors (diagram below just depicts a simpler version but it has many -many ) connecting with $A,B,C$ points by wires which may not be of same value, ( but its true that current is flowing in all wires in the small circuit  when battery is connected in full big circuit) ,how can we prove that it always can be converted to this form as shown in below figure by repeated Star-Delta transform where "O" is a point neecessary to make current flow in all  ? 
Note : Circle means many resistance with wires are there inside that . I am just asking general any sort of connections with other wires will always convert it into that form , if there is any exception please tell
 A: Well at the end of the day if electricity flow is coming in let's say by A and going out by B and C, you will have some proportion of how much goes through B and how much through C and that will give you the ratio of the resistances in the diagram below.
Then also with the measure of the total resistance of one of that two paths (measuring I and V) you will be able to exactly specify the exact values, so yeah, a circuit with 3 external legs can be drawn as an equivalent diagram as the picture below, which is easily seen by thinking only with flows.
In other words, you don't really care how the flow distributes through the middle steps, if you know the last ones (which you can measure easily), then you can write the simple equivalent diagram below that gives the same outputs:
Also, the resistance in front of A, given my answer one could think can be neglected, since you only need the resistances in B and C to get the proportions and modulus right, but if I recall correctly is necessary in order that both I (flow) and also V stay consistent. I suppose you can check that last statement :).
