Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O^*(\gamma^D)$ time for each constant $k$, where $\gamma$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.

翻译：在大图表中列出密度较大的子图在网络分析应用程序的种类中扮演着关键任务,比如社区检测。 clique, 作为最稠密的模型, 已被广泛调查。 但是, 在实际中, 社区由于各种原因, 例如数据噪音等, 很少形成 clique 。 因此, $- plex, -- 以每个顶点相邻, 但最多为 $k$ 的图表, 被引入一个宽松的 cluque 版本。 通常, 要更好地模拟具有凝聚力的社区, 重点是与小美元相联的 $- liver 。 在本文中, 我们继续列出所有最大 $- lex 和 最大 $- lex- level- 格式, 我们的第一个 listalPlexPlex, 列出每个常值$( gammamamamamamamama) 的最大值, $\ gramme- lax lex lex level- level- level of we more level level level legal level level level- legal level- level- level- level- levels the we mold level- level- level- levels ex ex ex ex level- ex ex ex legal ex ex ex ex level- level- level- legs ex ex ex ex level levels ex ex ex level- ex, ex ex ex ex excolmocucefors levels levels levels levels levels ex ex ex ex lex lexs ex ex ex exal levelss ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex ex