Glycerol on heating with anhydrous magnesium sulphate yield an acid smelling distillate of molecular formula of C3H4O which immediately decolouries bromine water and restore the colour to Schiff's reagent. Indicate a possible structure for the compound

The formula for DoU is $\text{C} + 1 - \frac{\text{H}}{2} - \frac{\text{X}}{2} + \frac{\text{N}}{2}$, where C is carbon, H is hydrogen, X is halogen, and N is nitrogen.
For $\text{C}_3\text{H}_4\text{O}$:

A DoU of 2 indicates the presence of two double bonds, one triple bond, or one ring and one double bond.

• "Immediately decolorizes bromine water" indicates the presence of a carbon-carbon double bond ($\text{C}=\text{C}$) or a triple bond ($\text{C}\equiv\text{C}$). This accounts for at least one degree of unsaturation.
• "Restore the colour to Schiff's reagent" indicates the presence of an aldehyde functional group ($\text{-CHO}$). An aldehyde group contains a carbon-oxygen double bond ($\text{C}=\text{O}$), which accounts for one degree of unsaturation.
• "Acid smelling distillate" refers to the pungent, acrid smell characteristic of small aldehydes.

Since the compound has an aldehyde group ($\text{C}=\text{O}$, 1 DoU) and a carbon-carbon double bond ($\text{C}=\text{C}$, 1 DoU), these two features together account for the total DoU of 2.
With 3 carbon atoms, we can arrange these groups as follows:
The aldehyde group is at one end of the carbon chain. The carbon-carbon double bond must be between the other two carbons.

The only possible arrangement for $\text{C}_3\text{H}_4\text{O}$ containing both a $\text{C}=\text{C}$ bond and an aldehyde ($\text{-CHO}$) group is prop-2-enal, also known as acrolein.
Its structure is $\text{CH}_2=\text{CH}-\text{CHO}$.
Let's check the formula: 3 carbons, 2+1+1 = 4 hydrogens, 1 oxygen. This matches $\text{C}_3\text{H}_4\text{O}$.
The reaction of glycerol with a dehydrating agent like anhydrous magnesium sulfate is known to produce acrolein.