Maximal cliques play a fundamental role in numerous application domains, where their enumeration can prove extremely useful. Yet their sheer number, even in sparse real-world graphs, can make them impractical to be exploited effectively. To address this issue, one approach is to enumerate $\ell$-isolated maximal cliques, whose vertices have (on average) less than $\ell$ edges toward the rest of the graph. By tuning parameter $\ell$, the degree of isolation can be controlled, and cliques that are overly connected to the outside are filtered out. Building on Tomita et al.'s very practical recursive algorithm for maximal clique enumeration, we propose four pruning heuristics, applicable individually or in combination, that discard recursive search branches that are guaranteed not to yield $\ell$-isolated maximal cliques. Besides proving correctness, we characterize both the pruning power and the computational cost of these heuristics, and we conduct an extensive experimental study comparing our methods with Tomita's baseline and with a state-of-the-art approach. Results show that two of our heuristics offer substantial efficiency improvements, especially on real-world graphs with social network properties.

翻译：极大团在众多应用领域中扮演着基础性角色，其枚举过程具有极高的实用价值。然而，即使在稀疏的现实世界图中，极大团的数量也可能极为庞大，导致其难以被有效利用。为解决这一问题，一种方法是枚举ℓ-孤立极大团，即团内顶点与图中其余部分之间的边数（平均）少于ℓ。通过调整参数ℓ，可以控制团的孤立程度，并过滤掉与外部过度连接的团。基于Tomita等人提出的极具实用性的极大团枚举递归算法，我们提出了四种剪枝启发式策略，这些策略可单独或组合应用，用于丢弃那些保证不会产生ℓ-孤立极大团的递归搜索分支。除了证明算法的正确性外，我们还分析了这些启发式策略的剪枝能力和计算成本，并通过广泛的实验研究，将我们的方法与Tomita的基线方法以及一种前沿方法进行了比较。结果表明，其中两种启发式策略显著提升了效率，尤其在具有社交网络特性的现实世界图上表现突出。