For a fixed integer $k\ge 2$, a $k$-community structure in an undirected graph is a partition of its vertex set into $k$ sets called communities, each of size at least two, such that every vertex of the graph has proportionally at least as many neighbours in its own community as in any other community. In this paper, we give a necessary and sufficient condition for a forest on $n$ vertices to admit a $k$-community structure. Furthermore, we provide an $\mathcal{O}(n^{2})$-time algorithm that computes such a $k$-community structure in a forest, if it exists. These results extend a result of [Bazgan et al., Structural and algorithmic properties of $2$-community structure, Algorithmica, 80(6):1890-1908, 2018]. We also show that if communities are allowed to have size one, then every forest with $n \geq k\geq 2$ vertices admits a $k$-community structure that can be found in time $\mathcal{O}(n^{2})$. We then consider threshold graphs and show that every connected threshold graph admits a $2$-community structure if and only if it is not isomorphic to a star; also if such a $2$-community structure exists, we explain how to obtain it in linear time. We further describe two infinite families of disconnected threshold graphs, containing exactly one isolated vertex, that do not admit any $2$-community structure. Finally, we present a new infinite family of connected graphs that may contain an even or an odd number of vertices without $2$-community structures, even if communities are allowed to have size one.

翻译：对于固定整数$k\ge 2$，无向图中的$k$-社区结构是指将顶点集划分为$k个称为社区的集合（每个社区至少包含两个顶点），使得图中每个顶点在其自身社区中的邻居比例至少不低于在其他任何社区中的邻居比例。本文给出了顶点数为$n$的森林存在$k$-社区结构的充要条件。此外，我们提出了一种$\mathcal{O}(n^{2})$时间复杂度的算法，用于在森林中计算这样的$k$-社区结构（如果存在）。这些结果推广了[Bazgan等人，《2-社区结构的结构与算法性质》，Algorithmica，80(6):1890-1908，2018]中的结论。我们还证明：若允许社区大小为1，则每个满足$n \geq k\geq 2$的森林都存在可在$\mathcal{O}(n^{2})$时间内找到的$k$-社区结构。接着考虑阈值图，证明每个连通阈值图存在$2$-社区结构当且仅当它不同构于星图；若存在这样的$2$-社区结构，我们给出在线性时间内获取该结构的方法。进一步描述了包含恰好一个孤立顶点的两类无穷不连通阈值图族，它们均不具有任何$2$-社区结构。最后，我们构造了一个新的无穷连通图族，其中可能包含偶数或奇数个顶点，且即使允许社区大小为1，这些图也不存在$2$-社区结构。