We introduce the notions of blackout-tolerant temporal α-spanner of a temporal graph G which is a subgraph of G that preserves the distances between pairs of vertices of interest in G up to a multiplicative factor of α, even when the graph edges at a single time-instant become unavailable. In particular, we consider the single-source, single-pair, and all-pairs cases and, for each case we look at three quality requirements: exact distances (i.e., α=1), almost-exact distances (i.e., α=1+ε for an arbitrarily small constant ε>0), and connectivity (i.e., unbounded α). We provide almost tight bounds on the size of such spanners for general temporal graphs and for temporal cliques, showing that they are either very sparse (i.e., they have O˜(n) edges) or they must have size Ω(n2) in the worst case, where n is the number of vertices of G. We also investigate multiple blackouts and k-edge fault-tolerant temporal spanners.

A temporal graph is an undirected graph $G=(V,E)$ along with a function that assigns a time-label to each edge in $E$. A path in $G$ with non-decreasing time-labels is called temporal path and the distance from $u$ to $v$ is the minimum length (i.e., the number of edges) of a temporal path from $u$ to $v$. A temporal $α$-spanner of $G$ is a (temporal) subgraph $H$ that preserves the distances between any pair of vertices in $V$, up to a multiplicative stretch factor of $α$. The size of $H$ is the number of its edges. In this work we study the size-stretch trade-offs of temporal spanners. We show that temporal cliques always admit a temporal $(2k-1)-$spanner with $\tilde{O}(kn^{1+\frac{1}{k}})$ edges, where $k>1$ is an integer parameter of choice. Choosing $k=\lfloor\log n\rfloor$, we obtain a temporal $O(\log n)$-spanner with $\tilde{O}(n)$ edges that has almost the same size (up to logarithmic factors) as the temporal spanner in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity. We then consider general temporal graphs. Since $Ω(n^2)$ edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that $\tilde{O}(n/\log(1+\varepsilon))$ edges suffice to obtain a stretch of $(1+\varepsilon)$, for any small $\varepsilon>0$. This result is essentially tight since there are temporal graphs for which any temporal subgraph preserving exact distances from a single-source must use $Ω(n^2)$ edges. We extend our analysis to prove an upper bound of $\tilde{O}(n^2/β)$ on the size of any temporal $β$-additive spanner, which is tight up to polylogarithmic factors. Finally, we investigate how the lifetime of $G$, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

Let $G$ be a directed graph with $n$ vertices and $m$ edges, and let $s \in V(G)$ be a designated source vertex. We consider the problem of single source reachability (SSR) from $s$ in presence of failures of edges (or vertices). Formally, a spanning subgraph $H$ of $G$ is a {\em $k$-Fault Tolerant Reachability Subgraph ($k$-FTRS)} if it has the following property. For any set $F$ of at most $k$ edges (or vertices) in $G$, and for any vertex $v\in V(G)$, the vertex $v$ is reachable from $s$ in $G-F$ if and only if it is reachable from $s$ in $H - F$. Baswana et.al. [STOC 2016, SICOMP 2018] showed that in the setting above, for any positive integer $k$, we can compute a $k$-FTRS with $2^k n$ edges. In this paper, we give a much simpler algorithm for computing a $k$-FTRS, and observe that it extends to higher connectivity as well. Our results follow from a simple application of \emph{important separators}, a well known technique in Parameterized Complexity.