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非均匀故障模型下的一种推广形式。具体而言，边被划分为"安全边"和"不安全边"，并假设故障仅影响不安全边。在无权重柔性图连通性问题中，给定一个无向图 $G = (V,E)$，其边集 $E$ 被划分为安全边集 $S$ 与不安全边集 $U$，任务是找到一个包含至多 $k$ 条边的边集 $T$，使得对于任意不安全边 $u \in T$，子图 $T - \{u\}$ 仍连通且覆盖所有顶点 $V$。该问题同时推广了生成树问题和哈密顿环问题。我们从固定参数可解性（FPT）角度对无权重柔性图连通性问题展开研究，展示了关于哪些参数可导致固定参数可解性、哪些参数会导致难解性的近乎完整二分结果。为此，我们分别针对顶点删除至团图距离和树宽度参数设计了FPT时间算法。通过利用与哈密顿环问题的紧密关联，我们证明在标准参数化复杂性假设下，针对许多更小参数设计FPT时间算法是不可能的。针对问题特定参数，我们发现当以不安全边数量为参数时，无权重柔性图连通性问题存在FPT时间算法。此外，我们研究了解决方案中边数的上界之下参数，证明该参数同样可引出FPT时间算法。