Counting small cohesive subgraphs in a graph is a fundamental operation with numerous applications in graph analysis. Previous studies on cohesive subgraph counting are mainly based on the clique model, which aim to count the number of $k$-cliques in a graph with a small $k$. However, the clique model often proves too restrictive for practical use. To address this issue, we investigate a new problem of counting cohesive subgraphs that adhere to the hereditary property. Here the hereditary property means that if a graph $G$ has a property $\mathcal{P}$, then any induced subgraph of $G$ also has a property $\mathcal{P}$. To count these hereditary cohesive subgraphs (\hcss), we propose a new listing-based framework called \hcslist, which employs a backtracking enumeration procedure to count all \hcss. A notable limitation of \hcslist is that it requires enumerating all \hcss, making it intractable for large and dense graphs due to the exponential growth in the number of \hcss with respect to graph size. To overcome this limitation, we propose a novel pivot-based framework called \hcspivot, which can count most \hcss in a combinatorial manner without explicitly listing them. Two additional noteworthy features of \hcspivot is its ability to (1) simultaneously count \hcss of any size and (2) simultaneously count \hcss for each vertex or each edge, while \hcslist is only capable of counting a specific size of \hcs and obtaining a total count of \hcss in a graph. We focus specifically on two \hcs: $s$-defective clique and $s$-plex, with several non-trivial pruning techniques to enhance the efficiency. We conduct extensive experiments on 8 large real-world graphs, and the results demonstrate the high efficiency and effectiveness of our solutions.

翻译：计数图中较小的凝聚子图是图分析中的基本操作，具有广泛的应用。以往关于凝聚子图计数的研究主要基于团模型，旨在计算图中小规模$k$-团的数目。然而，团模型在实际应用中往往过于严格。为解决此问题，我们研究了一个新问题：计数满足遗传性质的凝聚子图。这里，遗传性质指若图$G$具有性质$\mathcal{P}$，则$G$的任意导出子图也具有性质$\mathcal{P}$。为计数这些遗传凝聚子图（\hcss），我们提出一个新的基于列举的框架，称为\hcslist，它采用回溯枚举过程来统计所有\hcss。\hcslist的一个显著限制是需要枚举所有\hcss，这使得其在大规模稠密图上难以处理，因为\hcss的数量随图规模呈指数增长。为克服此限制，我们提出一种新颖的基于枢轴的框架，称为\hcspivot，它能以组合方式计数大部分\hcss，而无需显式列出它们。\hcspivot另外两个值得注意的特性是：（1）能同时计数任意大小的\hcss；（2）能为每个顶点或每条边同时计数\hcss，而\hcslist仅能计数特定大小的\hcs并获取图中\hcss的总数。我们具体关注两种\hcs：$s$-缺陷团和$s$-plex，并采用若干非平凡剪枝技术提升效率。我们在8个大型真实世界图上进行了广泛实验，结果证明了我们方案的高效性和有效性。