Which of the following are pairs of equivalent circuits

The lamp illuminates when the circuit path evaluates to logical '1'. The equivalence of circuits is determined by their simplified Boolean expressions, where series switches combine via AND and parallel via OR.

Circuit (A): The top branch has $S_1$ and $S_2$ in series, and the bottom branch has $S_1$ and $S_3$ in series, with both branches in parallel. The expression is $(S_1 \cdot S_2) + (S_1 \cdot S_3)$, which simplifies to $S_1 \cdot (S_2 + S_3)$ by the distributive property.

Circuit (B): $S_1$ and $S_2$ are in parallel, giving $S_1 + S_2$.

Circuit (C): $S_1$ in series with the parallel combination of $S_2$ and $S_3$, yielding $S_1 \cdot (S_2 + S_3)$.

Circuit (D): All switches in series: $S_1 \cdot S_2 \cdot S_3$.

Circuit (E): $S_3$ in parallel with the series of $S_1$ and $S_2$, giving $S_1 \cdot S_2 + S_3$.

Circuits (A) and (C) share the same simplified expression $S_1 \cdot (S_2 + S_3)$, confirming their equivalence.

Final answer: $\boxed{\text{D}}$