$k$-core is a subgraph where every node has at least $k$ neighbors within the subgraph. The $k$-core subgraphs has been employed in large platforms like Network Repository to comprehend the underlying structures and dynamics of the network. Existing studies have primarily focused on finding $k$-core groups without considering their size, despite the relevance of solution sizes in many real-world scenarios. This paper addresses this gap by introducing the size-prescribed $k$-core search (SPCS) problem, where the goal is to find a subgraph of a specified size that has the highest possible core number. We propose two algorithms, namely the {\it TSizeKcore-BU} and the {\it TSizeKcore-TD}, to identify cohesive subgraphs that satisfy both the $k$-core requirement and the size constraint. Our experimental results demonstrate the superiority of our approach in terms of solution quality and efficiency. The {\it TSizeKcore-BU} algorithm proves to be highly efficient in finding size-prescribed $k$-core subgraphs on large datasets, making it a favorable choice for such scenarios. On the other hand, the {\it TSizeKcore-TD} algorithm is better suited for small datasets where running time is less critical.

翻译：$k$-核是指子图中每个节点至少拥有$k$个邻居的子图。$k$-核子图已被应用于Network Repository等大型平台，以理解网络的底层结构和动态特性。现有研究主要关注寻找$k$-核群体，而忽略了它们的规模，尽管在许多实际场景中解的大小具有相关性。本文通过提出指定大小的$k$-核搜索（SPCS）问题来填补这一空白，其目标是找到指定规模且具有最大可能核数的子图。我们提出了两种算法，即{\it TSizeKcore-BU}和{\it TSizeKcore-TD}，用于识别同时满足$k$-核要求和规模约束的凝聚子图。实验结果表明，我们的方法在解的质量和效率方面具有优越性。{\it TSizeKcore-BU}算法在大型数据集上寻找指定大小的$k$-核子图方面表现出高效性，使其成为此类场景的优选方案。而{\it TSizeKcore-TD}算法则更适用于运行时间要求不高的小型数据集。