There seems to be a mistake in the given information. Potassium fumarate (C4H2K2O4) is not directly related to the production of water through electrolysis. Electrolysis is a chemical process in which an electric current is used to break down a compound into its constituent elements. In the case of water (H2O), electrolysis can be used to separate it into hydrogen gas (H2) and oxygen gas (O2).

However, if we assume that the given compound is actually potassium formate (HCOOK), then we can explain the answer. Potassium formate can be electrolyzed to produce water and carbon dioxide gas.

Here is an explanation of the process:

1. Electrolysis of Potassium Formate:
- When an electric current is passed through an aqueous solution of potassium formate, the compound undergoes electrolysis.
- At the anode (positive electrode), oxidation occurs, and oxygen gas is produced:
2HCOOK(aq) -> 2CO2(g) + 2H2O(l) + O2(g) + 4e^-
- At the cathode (negative electrode), reduction occurs, and hydrogen gas is produced:
4H2O(l) + 4e^- -> 4OH^-(aq) + 2H2(g)

2. Formation of Water:
- The reduction reaction at the cathode results in the formation of hydroxide ions (OH^-) and hydrogen gas (H2).
- The hydroxide ions can react with the hydrogen ions (H+) from the potassium formate to produce water (H2O):
4OH^-(aq) + 4H+(aq) -> 4H2O(l)

Therefore, the correct answer is option 'B' (C2H2) if the given compound is potassium formate (HCOOK). Electrolyzing potassium formate would result in the production of water (H2O) along with carbon dioxide gas (CO2).