  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.
