We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on $n$ vertices, it is polynomial-time solvable on graphs with minimum degree $n-3$ and NP-hard on $(n-4)$-regular graphs. We prove that it is polynomial-time $4/3$-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree $n-t$ for any constant $t\geq 4$.

翻译：我们考虑将一个图形分割成一个非固定数量的非重叠最大密度子图的问题。 分区的密度是子图密度的总和, 子图的密度是其平均程度, 即其边缘数和脊椎数的比率。 这个问题叫做“ 注意图分区 ”, 已知在普通图和树上可粘贴的多球时线上是NP硬的, 多球时2 准。 在本文中, 我们研究对特定稀有和稠密的图形类限制Dense 图片的密度。 特别是, 我们证明, 在稠密的双面图和立方图上是硬硬的。 在以美元为顶尖的密度图上, 它在最小值为 $-3 和 $- n-4 平面图上是多球时可粘固的。 我们证明, 在任何以美元为基数的 4/3美元正数的硬度正方位图上, 任何以美元为4/3美元的正值正数的正方位图上, 。