We study a generalization of the classic Spanning Tree problem that allows for a non-uniform failure model. More precisely, edges are either \emph{safe} or \emph{unsafe} and we assume that failures only affect unsafe edges. In Unweighted Flexible Graph Connectivity we are given an undirected graph $G = (V,E)$ in which the edge set $E$ is partitioned into a set $S$ of safe edges and a set $U$ of unsafe edges and the task is to find a set $T$ of at most $k$ edges such that $T - \{u\}$ is connected and spans $V$ for any unsafe edge $u \in T$. Unweighted Flexible Graph Connectivity generalizes both Spanning Tree and Hamiltonian Cycle. We study Unweighted Flexible Graph Connectivity in terms of fixed-parameter tractability (FPT). We show an almost complete dichotomy on which parameters lead to fixed-parameter tractability and which lead to hardness. To this end, we obtain FPT-time algorithms with respect to the vertex deletion distance to cluster graphs and with respect to the treewidth. By exploiting the close relationship to Hamiltonian Cycle, we show that FPT-time algorithms for many smaller parameters are unlikely under standard parameterized complexity assumptions. Regarding problem-specific parameters, we observe that Unweighted Flexible Graph Connectivity} admits an FPT-time algorithm when parameterized by the number of unsafe edges. Furthermore, we investigate a below-upper-bound parameter for the number of edges of a solution. We show that this parameter also leads to an FPT-time algorithm.

翻译：我们研究了经典生成树问题在允许非均匀故障模型下的推广。更精确地说，边被分为**安全边**和**不安全边**，且假设故障仅影响不安全边。在无权重灵活图连通性问题中，给定一个无向图 $G = (V,E)$，其边集 $E$ 被划分为安全边集 $S$ 和不安全边集 $U$，任务是在至多 $k$ 条边构成的集合 $T$ 中，使得对于任意不安全边 $u \in T$，$T - \{u\}$ 仍保持连通且覆盖 $V$。无权重灵活图连通性问题同时推广了生成树问题和哈密顿回路问题。我们从固定参数可解性（FPT）角度研究该问题，展示了关于哪些参数导致固定参数可解性、哪些导致难解性的近乎完全二分性划分。为此，我们针对顶点删除距离到簇图的参数以及树宽参数分别提出了FPT时间算法。通过利用与哈密顿回路问题的紧密关联，我们证明在标准参数化复杂度假设下，许多更小参数的FTP时间算法不太可能存在。针对问题特定参数，我们发现当以不安全边数量作为参数时，无权重灵活图连通性问题存在FPT时间算法。此外，我们研究了关于解中边数的"低于上界"参数，并证明该参数同样能导出FPT时间算法。