Let $T$ be a rooted tree in which a set $M$ of vertices are marked. The lowest common ancestor (LCA) of $M$ is the unique vertex $\ell$ with the following property: after failing (i.e., deleting) any single vertex $x$ from $T$, the root remains connected to $\ell$ if and only if it remains connected to some marked vertex. In this note, we introduce a generalized notion called $f$-fault-equivalent LCAs ($f$-FLCA), obtained by adapting the above view to $f$ failures for arbitrary $f \geq 1$. We show that there is a unique vertex set $M^* = \operatorname{FLCA}(M,f)$ of minimal size such after the failure of any $f$ vertices (or less), the root remains connected to some $v \in M$ iff it remains connected to some $u \in M^*$. Computing $M^*$ takes linear time. A bound of $|M^*| \leq 2^{f-1}$ always holds, regardless of $|M|$, and holds with equality for some choice of $T$ and $M$.

翻译：设$T$为一棵带标记顶点集$M$的有根树。$M$的最低公共祖先（LCA）是满足以下性质的唯一顶点$\ell$：当$T$中任意单个顶点$x$发生故障（即被删除）后，根节点与$\ell$保持连通当且仅当其与某个标记顶点保持连通。本文提出一种称为$f$故障等价最低公共祖先（$f$-FLCA）的广义概念，通过将上述视角推广至任意$f \geq 1$个故障的情形而获得。我们证明存在唯一的最小规模顶点集$M^* = \operatorname{FLCA}(M,f)$，使得在任意$f$个（或更少）顶点发生故障后，根节点与某个$v \in M$保持连通当且仅当其与某个$u \in M^*$保持连通。计算$M^*$仅需线性时间。无论$|M|$取值如何，始终满足$|M^*| \leq 2^{f-1}$的界，且对于特定的$T$和$M$选择，该界可达紧确。