  Temporal graphs represent interactions between entities over time. These
interactions may be direct, a contact between two vertices at some time
instant, or indirect, through sequences of contacts called journeys. Deciding
whether an entity can reach another through a journey is useful for various
applications in complex networks. In this paper, we present a disk-based data
structure that maintains temporal reachability information under the addition
of new contacts in a non-chronological order. It represents the \emph{timed
transitive closure} (TTC) by a set of \emph{expanded} R-tuples of the form $(u,
v, t^-, t^+)$, which encodes the existence of journeys from vertex $u$ to
vertex $v$ with departure at time $t^-$ and arrival at time $t^+$. Let $n$ be
the number of vertices and $\tau$ be the number of timestamps in the lifetime
of the temporal graph. Our data structure explicitly maintains this information
in linear arrays using $O(n^2\tau)$ space so that sequential accesses on disk
are prioritized. Furthermore, it adds a new unsorted contact $(u, v, t)$
accessing $O\left(\frac{n^2\tau}{B}\right)$ sequential pages in the worst-case,
where $B$ is the of pages on disk; it answers whether there is of a journey
from a vertex $u$ to a vertex $v$ within a time interval $[t_1, t_2]$ accessing
a single page; it answers whether all vertices can reach each other in $[t_1,
t_2]$; and it reconstructs a valid journey that validates the reachability from
a vertex $u$ to a vertex $v$ within $[t_1, t_1]$ accessing
$O\left(\frac{n\tau}{B}\right)$ pages. Our experiments show that our novel data
structure are better that the best known approach for the majority of cases
using synthetic and real world datasets.
