In the circuit shown in the figure all the resistance are identical and each has the value r$$\Omega $$. The equivalent resistance of the combination between the points A and B will remain unchanged even when the following pairs of points marked in the figure are connected through a resistance R.

A non-zero current passes through the galvanometer G shown in the circuit when the key K is closed and its value does not change when the key is opened. Then, which of the following statement(s) is/are true?