Exchange graphs and Ext-quivers of hearts of tube categories

In this paper we introduce the notion of pre-simple-minded collection (pre-SMC) of type $\mathbb{A}$ in the bounded derived categories $\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$ of tube categories $\textbf{T}_{p}$ of rank $p$. This provides an effective approach to classify the hearts in $\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$. We then use this classification to prove the exchange graph of hearts in $\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$ is connnected. Further, we classify the Ext-quivers of hearts in $\mathcal{D}^{b} (${\rm{\textbf{T}}}$_p)$. As an application, we show that the space of Bridgeland stability conditions on $\mathcal{D}^{b}(${\rm{\textbf{T}}}$_p)$ is connected.