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Dendric shifts are defined by combinatorial restrictions of the extensions ofthe words in their languages. This family generalizes well-known families ofshifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of intervalexchange transformations. It is known that any minimal dendric shift has aprimitive $\mathcal{S}$-adic representation where the morphisms in$\mathcal{S}$ are positive tame automorphisms of the free group generated bythe alphabet. In this paper we investigate those $\mathcal{S}$-adicrepresentations, heading towards an $\mathcal{S}$-adic characterization of thisfamily. We obtain such a characterization in the ternary case, involving adirected graph with 2 vertices.

$\mathcal{S}$-adic characterization of minimal ternary dendric shifts