CSP dichotomy for special triads

For a fixed digraph $\mathbb{G}$, the Constraint Satisfaction Problem with the template $\mathbb{G}$, or $\mathrm{CSP}(\mathbb{G})$ for short, is the problem of deciding whether a given input digraph $\mathbb{H}$ admits a homomorphism to $\mathbb{G}$. The dichotomy conjecture of Feder and Vardi states that $\mathrm{CSP}(\mathbb{G})$, for any choice of $\mathbb{G}$, is solvable in polynomial time or NP-complete. This paper confirms the conjecture for a class of oriented trees called special triads. As a corollary we get the smallest known example of an oriented tree (with $33$ vertices) defining an NP-complete $\mathrm{CSP}(\mathbb{G})$.