Coradically graded Hopf algebras of tame corepresentation type

Let $\Bbbk$ be an algebraically closed field of characteristic $0$ and let $H$ be a finite-dimensional Hopf algebra over $\Bbbk$ with the dual Chevalley property. In this paper, we give a description of the link quiver of $H$ for different corepresentation types. Moreover, we show that $\operatorname{gr}^c(H)$ is of tame corepresentation type if and only if $\operatorname{gr}^c(H)\cong (\k\langle x,y\rangle/I)^* \times H_0$ for some special ideals $I$. Using the methods of link quivers and bosonization, we then discuss which of the above ideals occur when $(\Bbbk\langle x,y\rangle/I)^* \times H_0$ is a Hopf algebra of tame corepresentation type under certain assumptions.