the discontinuity may occur along the trace, at nearby points. More precisely, we will look at a function $u:{\ensuremath{\mathbb{R}}}^2\to{\ensuremath{\mathbb{R}}}$, which is an $s$-minimal graph in $(-2,2)\times(0,4)$ and is such that $u=0$ in $ (-2,2)\times(-h,0)$, for some $h>0$. In this setting, we will consider its trace along $\{x_2=0\}$, namely we consider the function $$\lim_{x_2\searrow0}u(x_1,x_2).$$ The main question that we address in this article is precisely whether or not [*the trace of a nonlocal minimal graph possesses any distinctive feature or satisfies any particular geometric constraint*]{}.

We will prove that, differently from the case of the linear equations (and also in sharp contrast to the case of classical minimal surfaces), the traces of nonlocal minimal graphs [*cannot have an arbitrary shape*]{}, and, in fact, [*matching points from the two sides must necessarily occur with horizontal tangencies*]{}.

This result relies on a classification theory for homogeneous graphs, since we will show that, in this case, [*the matching at the origin is sufficient to make a nonlocal minimal graph trivial*]{}.

In the rest of this introduction, we will present the precise mathematical framework in which we work and provide the formal statements of our main results.

Boundary flexibility of linear fractional equations
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We 