The 2n vertices of a graph G correspond to all subsets of a set of size n, for…

Concept: In this intersection graph, a vertex is a subset A of the n-element set, and its degree equals the number of other subsets B with |A ∩ B| = 2. Since an intersection can never be larger than the smaller of the two sets, |A ∩ B| can reach 2 only when |A| ≥ 2 — so any subset of size 0 or 1 is automatically degree zero, while any subset of size ≥ 2 must be checked for whether such a B actually exists.

Application:

1. Size 0 and size 1 subsets: the empty set intersects everything in 0 elements, and each 1-element subset intersects any other set in at most 1 element — neither can ever reach an intersection of size 2, so all of these vertices are degree zero. Count: 1 (empty set) + n (singletons) = n + 1.

2. Size-≥2 subsets always have a partner: take any subset A with |A| ≥ 2 and pick two of its elements {a, b}. If |A| < n, let B = {a, b} together with one element outside A (n ≥ 6 guarantees such an element exists); then A ∩ B = {a, b}, size 2, and B ≠ A. If A is the full n-element set, B = {a, b} itself is a distinct subset with A ∩ B = {a, b}. Either way A has a neighbor at intersection exactly 2, so A is not degree zero.

3. This is exactly what resolves a common doubt: a 2-element subset such as {1, 2} is not isolated just because it is disjoint from other 2-element subsets such as {3, 4} — it only needs ONE subset that intersects it in exactly two elements, and the full n-element set (or any 3-element superset containing it) always supplies that neighbor.

Cross-check: with n = 6 (the smallest allowed value), there are C(6,2) = 15 subsets of size 2, and each intersects the full 6-element set in exactly itself (size 2) — confirming none of them is degree zero. The empty set plus the 6 singletons give 7 degree-zero vertices, matching n + 1 = 7.

Result: the number of degree-zero vertices is n + 1.