Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[S_i] \in \mathcal{H}_i$, $G[S_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We show that, in some known cases, the obtained running times are comparable to those of the best know algorithms.

翻译：给定两个整数 $\ell$ 和 $p$ 以及 $\ell$ 个图类 $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$，问题 $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$、$\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ 和 $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ 分别询问：输入图 $G$ 能否将 $V(G)$、$V(G)$、$E(G)$ 划分为 $\ell$ 个集合 $S_1, \ldots, S_\ell$，使得对于每个 $i$ 介于 $1$ 到 $\ell$ 之间，分别有 $G[S_i] \in \mathcal{H}_i$、$G[S_i] \in \mathcal{H}_i$、$(V(G),S_i) \in \mathcal{H}_i$。此外，在 $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$ 中，我们要求端点位于划分中不同集合的边的数量不超过 $p$。我们证明：若存在基于动态规划树分解的算法来识别每个图类 $\mathcal{H}_i$，则我们可以构造性地创建基于动态规划树分解的算法，用于求解 $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$、$\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ 和 $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$。我们表明，在某些已知情形下，所得运行时间与现有最优算法相当。