How many cyclic structures are possible for $$\text{C}_4\text{H}_6$$?

The degree of unsaturation (also called index of hydrogen deficiency, IHD) tells us how many $$\pi$$-bonds and/or rings are present in a hydrocarbon.

For a general hydrocarbon $$C_nH_m$$, $$\text{IHD}= \dfrac{2n+2-m}{2}$$.

For $$C_4H_6$$, $$\text{IHD}= \dfrac{2(4)+2-6}{2}= \dfrac{10-6}{2}=2$$.
Thus every valid structure must contain a total of two units of unsaturation. Each ring counts as 1 unit and each double bond also counts as 1 unit.

Because the question asks for cyclic structures, at least one ring must be present. Therefore we have two possible patterns of unsaturation:
Case 1: one ring + one double bond.
Case 2: two rings and no double bond (bicyclic system).

Case 1: One ring + one double bond (four-carbon skeleton)

1. Cyclobutene : $$\mathrm{C_4H_6}$$ with the double bond inside the 4-membered ring.
Only one constitutional arrangement is possible because all C atoms of cyclobutene are equivalent.

2. Methyl-substituted cyclopropenes (3-membered ring contains the double bond).     a) 1-Methylcyclopropene (methyl on a vinylic carbon, unique because the two vinylic positions are equivalent).     b) 3-Methylcyclopropene (methyl on the saturated ring carbon).     These two are not identical, so they give two distinct structures.

3. Methylenecyclopropane : a 3-membered ring where the double bond is exocyclic, i.e. $$\mathrm{CH_2{=}C_3}$$ (ring carbon 3 is doubly bonded to an external $$\mathrm{CH_2}$$ group).     This satisfies one ring + one double bond and is different from the cyclopropenes above.

Hence, under Case 1 we have 1 (cyclobutene) + 2 (methylcyclopropenes) + 1 (methylenecyclopropane) = 4 distinct cyclic structures.

Case 2: Two rings, no double bond

With four carbons and two rings we need five single C-C bonds (because $$\text{rings} = \text{bonds} - n + 1$$). The unique way to connect four carbon atoms with five single bonds is the “edge-fused” bicyclic skeleton $$\mathrm{bicyclo[1.1.0]butane}$$ (two cyclopropane rings sharing one edge). This gives one additional structure.

Total count

Adding Case 1 (4 structures) and Case 2 (1 structure) gives $$4+1 = 5$$ cyclic structural isomers for $$\mathrm{C_4H_6}$$.

Option B which is: 5