In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $λ$ and $k$. The task is to insert at most $k$ links from $L$ to $G$ to make $G$ $λ$-vertex-connected. We show that the problem is fixed-parameter tractable (FPT) when parameterized by $λ$ and $k$, by giving an algorithm with running time $2^{O(k \log (k + λ))} n^{O(1)}$. This improves upon a recent result of Carmesin and Ramanujan [SODA'26], who showed that the problem is FPT parameterized by $k$ but only when $λ\le 4$. We also consider the analogous edge connectivity augmentation problem, where the goal is to make $G$ $λ$-edge-connected. We show that the problem is FPT when parameterized by $k$ only, by giving an algorithm with running time $2^{O(k \log k)} n^{O(1)}$. Previously, such results were known only under additional assumptions on the edge connectivity of $G$.

翻译：在顶点连通性增强问题中，给定一个无向 $n$ 顶点图 $G$、一个链接集合 $L \subseteq \binom{V(G)}{2} \setminus E(G)$，以及整数 $λ$ 和 $k$。任务是从 $L$ 中向 $G$ 插入至多 $k$ 条链接，使得 $G$ 成为 $λ$-顶点连通的。我们证明了该问题在参数化为 $λ$ 和 $k$ 时是固定参数可解的 (FPT)，并给出一个运行时间为 $2^{O(k \log (k + λ))} n^{O(1)}$ 的算法。这改进了 Carmesin 和 Ramanujan 近期的一个结果 [SODA'26]，他们证明了当 $λ\le 4$ 时该问题在参数化为 $k$ 时是 FPT 的。我们还考虑了类似的边连通性增强问题，其目标是使 $G$ 成为 $λ$-边连通的。我们证明了该问题在仅参数化为 $k$ 时是 FPT 的，并给出一个运行时间为 $2^{O(k \log k)} n^{O(1)}$ 的算法。此前，这类结果仅在对 $G$ 的边连通性有额外假设时成立。