This article introduces the Generalized Minimum Branch Vertices problem. Given an undirected graph, where the set of vertices is partitioned into clusters, the Generalized Minimum Branch Vertices problem consists of finding a tree spanning exactly one vertex for each cluster and having the minimum number of branch vertices, namely vertices with degree greater than two. When each cluster is a singleton, the problem reduces to the well-known Minimum Branch Vertices problem, which is NP-hard. We show some properties that any feasible solution to the problem has to satisfy. Some of these properties can be used to determine useless vertices or edges, which can be removed to reduce the size of the instances. We propose an integer linear programming formulation for the problem, we derive the dimension of the polytope, we study the trivial inequalities and introduce two new classes of valid inequalities, that are proved to be facet-defining.

中文翻译：

---

广义最小分支顶点问题：性质和多面体分析

本文介绍了广义最小分支顶点问题。给定一个无向图，其中顶点集被划分为集群，广义最小分支顶点问题包括为每个集群找到一棵树，该树恰好跨越一个顶点，并且具有最少数量的分支顶点，即度数大于 2 的顶点。当每个集群都是单例时，问题就简化为众所周知的最小分支顶点问题，这是 NP-hard 问题。我们展示了问题的任何可行解决方案必须满足的一些属性。其中一些属性可用于确定无用的顶点或边，可以将其删除以减小实例的大小。我们为这个问题提出了一个整数线性规划公式，我们推导出多面体的维数，