In search of the lost tree: Hardness and relaxation of spanning trees in temporal graphs

A graph whose edges only appear at certain points in time is called a temporal graph (among other names). These graphs are temporally connected if all ordered pairs of vertices are connected by a path that traverses edges in chronological order (a temporal path). Reachability in temporal graphs departs significantly from standard reachability; in particular, it is not transitive, with structural and algorithmic consequences. For instance, temporally connected graphs do not always admit spanning trees, i.e., subsets of edges that form a tree and preserve temporal connectivity among the nodes. In this paper, we revisit fundamental questions about the loss of universality of spanning trees. To start, we show that deciding if a spanning tree exists in a given temporal graph is NP-complete. What could be appropriate replacement for the concept? Beyond having minimum size, spanning trees enjoy the feature of enabling reachability along the same underlying paths in both directions, a pretty uncommon feature in temporal graphs. We explore relaxations in this direction and show that testing the existence of bidirectional spanning structures (bi-spanners) is tractable in general. On the down side, finding \emph{minimum} such structures is NP-hard even in simple temporal graphs. Still, the fact that bidirectionality can be tested efficiently may find applications, e.g. for routing and security, and the corresponding primitive that we introduce in the algorithm may be of independent interest.