Concepts/Formula used:
Resistors in Series:

Resistors in parallel:

(a)

The resistors across dc and cb are in series.

Now, the two resistors across db are in parallel.

The resistor across db and ad are in series.

The circuit can be redrawn as follows:

The resistances a’b’ and ab are in parallel.

(b)

The point b is connecting the ends of three resistors; the other ends of the resistors are attached to a fourth one with one of its end being point a.

Hence, we can redraw the circuit as follows:

We can see that the resistors across bc are in parallel.

We can redraw the circuit as follows:

Note that the resistors across bc and ca are in series.

(c) We can redraw the given circuit as follows:

Note that there is symmetry in the circuit; the upper and the lower parts are identical. Hence, the potential at d should be the same as potential at c. Consequently, there is no potential difference across dc. We can neglect the resistors across dc.

We can redraw this circuit as follows:

We can easily see that uc and cv are in series.

Similarly,

Now, resistors across uv and nm are in parallel.

(d)

Note that one end of all four resistors is connected to one point a. The other ends are connected to a circle with no resistance. Hence, all points in the circle are same i.e. point b. Thus, all the other ends are connected to the same point b. This simply describes resistors in parallel.

(e)

The circuit can be drawn as follows:

As we wish to find the resistance between points a and b, we have proceeded to add a voltage source of emf ϵ between the points a and b.

Let the net current coming out of the battery be 2I. We can see that the circuit is symmetric i.e. its upper and lower portion are the identical. Thus, the current should divide equally when branching out. So, current through ac and ad is I.

Now,

Hence,

As there is no potential difference across dc, there is no current passing through dc.

Hence, we can rewrite the circuit without the resistor across dc.

The circuit can be redrawn as:

We can see that uc and cv are in series.

Also, nd and dm are is series.

Hence, we can rewrite the circuit as follows:

Now, uv and nm are in parallel.