Abstract
A temporal digraph ${\cal G}$ 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 $uv\in E(G)$ when $u$ and $v$ are linked. Given a static digraph $G$, and a subset $R\subseteq V(G)$, a spanning branching with root $R$ is a subdigraph of $G$ that has exactly one path from $R$ to each $v\in V(G)$. In this paper, we consider the temporal version of Edmonds' classical result about the problem of finding $k$ edge-disjoint spanning branchings respectively rooted in given $R_1,\cdots,R_k$. We introduce and investigate different definitions of spanning branchings, and of edge-disjointness in the context of temporal digraphs. A branching ${\cal B}$ 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 each $v$ can be reached from the root at every time where $v$ is active. On the other hand, two branchings ${\cal B}_1$ and ${\cal B}_2$ 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.
Highlights
A temporal digraph is a digraph that changes in time
If a vertex v ∈ V (G) is active at every t ∈ T, we say that v is permanent
Given a temporal digraph G with base digraph G, and subsets of vertices in time R1, · · ·, Rk, i.e., sets of pairs (u, t) where u is a vertex of G and t a time stamp, here we investigate the many variations of finding disjoint spanning branchings with roots R1, · · ·, Rk
Summary
A temporal digraph is a digraph that changes in time. That is, given a static digraph G, a temporal digraph G with base static digraph G and lifetime set T changes as follows: at each time stamp t ∈ T , only a subdigraph of G is active, and edges might have a delay, leaving a vertex at some time stamp but arriving only later. If vertices are permanent or G has lifetime 2, edge-disjoint temporal-spanning can be solved in polynomial time, and in linear time on the size of G if all edges have a delay or if each snapshot is a DAG. When vertices can be inactive, we have to further ensure that, when waiting for the edge on a certain vertex, it must remain active in the waiting period [8] In this scenario, the definitions of reachability and connectivity change and it is natural to ask how well-known structures and results from graph theory in the classical sense change taking into account the temporal constraint.
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