A quiver presentation of the type D arc algebras and intrinsic formality of ${\mathbb{D}}_4$

We construct a presentation of the Khovanov arc algebras of type $D$ by a quiver with quadratic relations. As an application, we use this presentation and the Koszul dual to compute some Hochschild cohomology groups of ${\mathbb{D}}_4$, and show that ${\mathbb{D}}_4$ is intrinsically formal, that is, it admits no non-trivial $A_{\mathrm{\infty }}$-deformations.