Step 1: Understanding the Concept:
Reduction involves a decrease in the oxidation state of an element. We need to calculate the oxidation states of the central atoms in the reactants and products for each reaction.
Step 2: Detailed Explanation:
(A) $Mg$ ($0$) to $Mg^{2+}$ ($+2$): Oxidation.
(B) In $HgCl_2$, $Hg$ is in the $+2$ state. In $Hg_2Cl_2$, the $Hg$ atoms are in the $+1$ state ($2x + 2(-1) = 0 \implies x = +1$). This is a reduction from $+2$ to $+1$.
(C) $C$ in $CH_4$ is $-4$, in $CO_2$ it is $+4$: Oxidation.
(D) $Fe$ in $[Fe(CN)_6]^{4-}$ is $+2$, in $[Fe(CN)_6]^{3-}$ it is $+3$: Oxidation.
(E) $S$ in $H_2S$ is $-2$, in $S$ it is $0$: Oxidation.
Step 3: Final Answer:
In reaction (B), mercury is reduced from the $+2$ state to the $+1$ state.