7 research outputs found
Faster Exploration of Degree-Bounded Temporal Graphs
A temporal graph can be viewed as a sequence of static graphs indexed by discrete time steps. The
vertex set of each graph in the sequence remains the same; however, the edge sets are allowed to
differ. A natural problem on temporal graphs is the Temporal Exploration problem (TEXP):
given, as input, a temporal graph G of order n, we are tasked with computing an exploration
schedule (i.e., a temporal walk that visits all vertices in G), such that the time step at which the
walk arrives at the last unvisited vertex is minimised (we refer to this time step as the arrival
time). It can be easily shown that general temporal graphs admit exploration schedules with
arrival time no greater than O(n^2). Moreover, it has been shown previously that there exists
an infinite family of temporal graphs for which any exploration schedule has arrival time Ω(n^2),
making these bounds tight for general TEXP instances. We consider restricted instances of
TEXP, in which the temporal graph given as input is, in every time step, of maximum degree d;
we show an O(n^2/log n) bound on the arrival time when d is constant, and an O(d log d · n^2 / log n) bound
when d is given as some function of n