Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, \break $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, 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 apply this approach to known problems. For well-studied problems, like VERTEX COVER and GRAPH $q$-COLORING, we obtain running times that are comparable to those of the best known problem-specific algorithms. For an exotic problem from bioinformatics, called DISPLAYGRAPH, this approach improves the known algorithm parameterized by treewidth.

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：对于输入图$G$，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$i$（$1 \leq i \leq \ell$），分别满足$G[S_i] \in \mathcal{H}_i$、$G[S_i] \in \mathcal{H}_i$、$(V(G),S_i) \in \mathcal{H}_i$。此外，在$\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$中，我们要求端点位于不同划分集合中的边数不超过$p$。我们证明：如果对于每个$\mathcal{H}_i$存在基于动态规划树分解的识别算法，那么我们可以构造性地为$\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$、$\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$和$\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$设计基于动态规划树分解的算法。我们将该方法应用于已知问题。对于经典问题（如VERTEX COVER和GRAPH $q$-COLORING），我们获得的运行时间与已知最优专用算法相当。对于生物信息学中的特殊问题DISPLAYGRAPH，该方法改进了已知的关于树宽的参数化算法。