  We focus on strongly connected, strong for short, digraphs since in this
setting distance is defined for every pair of vertices.
  Distance ideals generalize the spectrum and Smith normal form of several
distance matrices associated with strong digraphs.
  We introduce the concept of pattern which allow us to characterize the family
$\Gamma_1$ of digraphs with only one trivial distance ideal over ${\mathbb Z}$.
  This result generalizes an analogous result for undirected graphs that states
that connected graphs with one trivial ideal over $\mathbb{Z}$ consists of
either complete graphs or complete bipartite graphs.
  It turns out that the strong digraphs in $\Gamma_1$ consists in the circuit
with 3 vertices and a family $\Lambda$ of strong digraphs that contains
complete graphs and complete bipartite graphs, regarded as digraphs. We also
compute all distance ideals of some strong digraphs in the family $\Lambda$.
  Then, we explore circuits, which turn out to be an infinite family of minimal
forbidden digraphs, as induced subdigraphs, for the strong digraphs in
$\Gamma_1$.
  This suggests that a characterization of $\Gamma_1$ in terms of forbidden
induced subdigraphs is harder than using patterns.
