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 $O(n^{2})$-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 $O(n^{2})$. 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$k$-社区结构是一种将顶点集划分为$k$个称为社区的集合（每个社区至少包含两个顶点）的划分，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其在其他任何社区中的邻居比例。本文给出了包含$n$个顶点的森林存在$k$-社区结构的充要条件。进一步地，我们提出一个$O(n^{2})$时间复杂度的算法，当该结构存在时可在森林中计算得到。这些结果扩展了[Bazgan等，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$O(n^{2})$时间内找到的$k$-社区结构。随后我们考虑阈值图，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；同时说明当此类2-社区结构存在时，如何在线性时间内获取。我们进一步描述了两种包含恰好一个孤立顶点的非连通阈值图无限族，它们不存在任何2-社区结构。最后，我们给出一个包含偶数或奇数个顶点且不含2-社区结构的新连通图无限族——即使允许社区大小为1的情况下亦然。