Given a graph G and a digraph D whose vertices are the edges of G, we investigate the problem of finding a spanning tree of G that satisfies the constraints imposed by D. The restrictions to add an edge in the tree depend on its neighborhood in D. Here, we generalize previously investigated problems by also considering as input functions \(\ell \) and u on E(G) that give a lower and an upper bound, respectively, on the number of constraints that must be satisfied by each edge. The produced feasibility problem is denoted by G-DCST, while the optimization problem is denoted by G-DCMST. We show that G-DCST is \(\texttt {NP}\)-complete even if functions \(\ell \) and u are taken under tight assumptions, as well as G and D. On the positive side, we prove two polynomial results, one for G-DCST and another for G-DCMST, and also give a simple exponential-time algorithm along with a proof that it is asymptotically optimal under the ETH. Finally, we prove that other previously studied constrained spanning tree (CST) problems can be modeled within our framework, namely, the Conflict CST, the Forcing CST, the At Least One/All Dependency CST, the Maximum Degree CST, the Minimum Degree CST, and the Fixed-Leaves Minimum Degree CST.

给定一个图G和一个顶点D为G的顶点的D图，我们研究寻找满足D约束的G生成树的问题。在树中添加边的限制取决于其在D中的邻域。在这里，我们通过考虑输入函数\（\ ell \）和E（G）上的u来概括先前研究的问题，它们分别给出了每个边必须满足的约束数量的下限和上限。产生的可行性问题用G-DCST表示，而优化问题用G-DCMST表示。我们证明，即使函数\（\ ell \）和u以及G和D都采用严格的假设，G-DCST也是\（\ texttt {NP} \）-完全的。在积极方面，我们证明了两个多项式结果，一个针对G-DCST，另一个针对G-DCMST，并且给出了一个简单的指数时间算法，并证明了它在ETH下是渐近最优的。最后，我们证明可以在我们的框架内对其他先前研究的约束生成树（CST）问题进行建模，即冲突CST，强制CST，至少一/全依赖CST，最大度CST，最小度CST和固定叶最小度CST。