A problem with the equivalent form of Frankl conjecture.

The Union Closed Sets Conjecture (or Frankl conjecture) is described in this link: https://en.wikipedia.org/wiki/Union-closed_sets_conjecture.

It is known that this problem has an equivalent form in terms of graphs.

Namely: Frankl conjecture is equivalent to assertion that in any graph $$G=(V,\mathcal{E})$$, such $$|\mathcal{E}|\ge 1$$, there exists two adjacent vertices that each of them belongs to at most half of the maximal independent sets.

But here is the problem:

Imagine the following graph $$G=(V,\mathcal{E})$$. Where :

$$V = \{1,2,3\}$$ and $$\mathcal{E}=\{\{1,2\},\{2,3\}\}$$

The only maximal independent set is $$\{1,3\}$$

Let's take any two vertices, for instance $$1,2$$. We see that $$1\in\{1,3\}$$ so "$$1$$" belongs to more than half of the maximal independent sets. That contradicts the conjecture.

However i strongly believe that i made some mistakes.

Where is mistake?

$$\{2\}$$ is also a maximal independent set - no vertices can be added to it without creating an edge.