  Positroids are families of matroids introduced by Postnikov in the study of
non-negative Grassmannians. In particular, positroids enumerate a CW
decomposition of the totally non-negative Grassmannian. Furthermore, Postnikov
has identified several families of combinatorial objects in bijections with
positroids. We will provide yet another characterization of positroids for
Gr$_{\geq 0}(2,n)$, the Grassmannians of lines, in terms of certain graphs. We
use this characterization to compute the dimension and the boundary of
positroid cells. This also leads to a combinatorial description of the
intersection of positroid cells, that is easily computable. Our techniques rely
on determining different ways to enlarge a given collection of subsets of
$\{1,\ldots,n\}$ to represent the dependent sets of a positroid, that is the
dependencies among the columns of a matrix with non-negative maximal minors.
Furthermore, we provide an algorithm to compute all the maximal positroids
contained in a set.
