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 present a necessary and sufficient condition for a forest $F$ to admit a $k$-community structure, for any integer $k\ge 2$. Furthermore, if such a $k$-community exists, it can be found in polynomial time. This generalises a result of Bazgan et al. (2018), who showed that all trees of size at least four, except stars, admit a $2$-community that can be found in polynomial time. We also show that, if communities are allowed to have size one, then every forest with at least $k\geq 2$ vertices admits a $k$-community structure that can be found in polynomial time. We then consider threshold graphs and show that every such connected 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, it can be found in polynomial time. Finally, we introduce a new infinite family of connected graphs that do not admit any $2$-community structure (even if communities are allowed to have size one). Such a family was presented in Bazgan et al. (2020), but its graphs all contained an even number of vertices. The graphs in our new family may contain an even or an odd number of vertices.
翻译:对于固定整数 $k\ge 2 美元, 一个非方向图形中的美元社区结构是其顶点分成为$k美元(所谓的社区)的分区, 每个大小至少为两张, 这样图中每个顶点的大小都与本社区中至少许多邻居的比例一样。 在本文中, 我们为森林输入一个美元社区结构提供了必要和充分的条件, 对于任何整数为$k\ge 2 美元。 此外, 如果存在一个 美元社区, 它可以在聚度时间中找到它的顶点。 这个概观是巴兹甘等人( 2018年) 的大小, 他显示所有大小的树至少四个, 除了星星, 都接受一个在聚度时间中能找到的$2美元社区。 我们还表明, 如果允许森林有1个大小, 那么每个森林含有至少$kgeq 2 的顶点, 也承认一个美元社区结构, 但是在聚度时间里可以找到一个 $k美元 。 we the contreal commal is a $ $ 2 coloral lium a listal listal is a listal.