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(B) To find the equivalent resistance between points $A$ and $B$,we analyze the circuit structure.$1$. The circuit consists of a resistor of $\frac{2r

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$2 r$

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(B) To find the equivalent resistance between points $A$ and $B$,we analyze the circuit structure.$1$. The circuit consists of a resistor of $\frac{2r}{3}$ in series with a parallel network.$2$. The parallel network consists of two branches. By simplifying the bridge-like structure using nodal analysis or symmetry,we find that the parallel part reduces to an equivalent resistance of $\frac{4r}{3}$.$3$. The total effective resistance $R_{	ext{net}}$ is the sum of the series resistor and the equivalent resistance of the parallel part:$R_{	ext{net}} = \frac{2r}{3} + \frac{4r}{3} = \frac{6r}{3} = 2r$.Thus,the correct option is $(b)$.

The effective resistance of the network between points $A$ and $B$ is

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