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{$\mathbf{\text{T}}$}}${}_p)$ of tube categories ${\mathbf{\text{T}}}_p$ of rank $p$. This provides an effective approach to classify the hearts in ${\mathcal{D}}^b(${\rm{$\mathbf{\text{T}}$}}${}_p)$. We then use this classification to prove the exchange graph of hearts in ${\mathcal{D}}^b(${\rm{$\mathbf{\text{T}}$}}${}_p)$ is connnected. Further, we classify the Ext-quivers of hearts in ${\mathcal{D}}^b(${\rm{$\mathbf{\text{T}}$}}${}_p)$. As an application, we show that the space of Bridgeland stability conditions on ${\mathcal{D}}^b(${\rm{$\mathbf{\text{T}}$}}${}_p)$ is connected.