Determine the equivalent resistance between the points A and B for...

The given circuit contains a diode in the upper branch. The diode's anode is connected towards terminal $A$ and its cathode towards terminal $B$.

If $V_A > V_B$, the diode is forward-biased and acts as a short circuit (ideal diode). The circuit consists of two $10\ \Omega$ resistors connected in parallel.
The equivalent resistance is $R_{eq} = \dfrac{10 \times 10}{10 + 10} = 5\ \Omega$.

If $V_A < V_B$, the diode is reverse-biased and acts as an open circuit. No current flows through the upper branch.
The equivalent resistance is simply the resistance of the lower branch, which is $R_{eq} = 10\ \Omega$.

Therefore, the equivalent resistance is $5\ \Omega$ if $V_A > V_B$ and $10\ \Omega$ if $V_A < V_B$.