A set of vertices in a graph forms a potential maximal clique if there exists a minimal chordal completion in which it is a maximal clique. Potential maximal cliques were first introduced as a key tool to obtain an efficient, though exponential-time algorithm to compute the treewidth of a graph. As a byproduct, this allowed to compute the treewidth of various graph classes in polynomial time. In recent years, the concept of potential maximal cliques regained interest as it proved to be useful for a handful of graph algorithmic problems. In particular, it turned out to be a key tool to obtain a polynomial time algorithm for computing maximum weight independent sets in $P_5$-free and $P_6$-free graphs (Lokshtanov et al., SODA `14 and Grzeskik et al., SODA `19. In most of their applications, obtaining all the potential maximal cliques constitutes an algorithmic bottleneck, thus motivating the question of how to efficiently enumerate all the potential maximal cliques in a graph $G$. The state-of-the-art algorithm by Bouchitt\'e \& Todinca can enumerate potential maximal cliques in output-polynomial time by using exponential space, a significant limitation for the size of feasible instances. In this paper, we revisit this algorithm and design an enumeration algorithm that preserves an output-polynomial time complexity while only requiring polynomial space.

翻译：图中的顶点集称为势极大团，若存在一个极小弦完全图在该图中该顶点集为极大团。势极大团最初作为关键工具被引入，用于获得计算图的树宽的高效（尽管是指数时间）算法。作为副产品，该方法允许在多项式时间内计算多种图类的树宽。近年来，势极大团的概念因被证明对多个图算法问题具有实用性而重新引起关注。特别地，它成为在 $P_5$ 自由图和 $P_6$ 自由图中获得最大权重独立集多项式时间算法的关键工具（Lokshtanov 等，SODA `14；Grzeskik 等，SODA `19）。在其大多数应用中，获取所有势极大团构成算法瓶颈，因此如何高效枚举图 $G$ 中所有势极大团成为重要问题。Bouchitt\'e 与 Todinca 的最新算法可在输出多项式时间内枚举势极大团，但需使用指数空间，这严重限制了可行实例的规模。本文重新审视该算法，设计了一种保持输出多项式时间复杂度的枚举算法，且仅需多项式空间。