Edge-Disjoint Branchings in Temporal Graphs

Abstract

A temporal digraph mathcal{G} is a triple (G, gamma , lambda ) where G is a digraph, gamma is a function on V(G) that tells us the time stamps when a vertex is active, and lambda is a function on E(G) that tells for each uvin E(G) when u and v are linked. Given a static digraph G, and a subset Rsubseteq V(G), a spanning branching with root R is a subdigraph of G that has exactly one path from R to each vin V(G). In this paper, we consider the temporal version of Edmonds’ classical result about the problem of finding k edge-disjoint spanning branchings respectively rooted at given R_1,cdots ,R_k. We introduce and investigate different definitions of spanning branchings, and of edge-disjointness in the context of temporal graphs. A branching mathcal{B} is vertex-spanning if the root is able to reach each vertex v of G at some time where v is active, while it is temporal-spanning if v can be reached from the root at every time where v is active. On the other hand, two branchings mathcal{B}_1 and mathcal{B}_2 are edge-disjoint if they do not use the same edge of G, and are temporal-edge-disjoint if they can use the same edge of G but at different times. This lead us to four definitions of disjoint spanning branchings and we prove that, unlike the static case, only one of these can be computed in polynomial time, namely the temporal-edge-disjoint temporal-spanning branchings problem, while the other versions are mathsf {NP}-complete, even under very strict assumptions.