EdgeDisjoint Branchings in Temporal Graphs
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 timestamps 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 $k$ edgedisjoint spanning branchings respectively rooted at given $R_1,\cdots,R_k$. We introduce and investigate different definitions of spanning branchings, and of edgedisjointness in the context of temporal graphs. A branching ${\cal B}$ is vertexspanning if the root is able to reach each vertex $v$ of $G$ at some time where $v$ is active, while it is temporalspanning if $v$ 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 edgedisjoint if they do not use the same edge of $G$, and are temporaledgedisjoint 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 temporaledgedisjoint temporalspanning branchings problem, while the other versions are $\mathsf{NP}$complete, even under very strict assumptions.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.12694
 Bibcode:
 2020arXiv200212694C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 05C85
 EPrint:
 16 pages, 4 figures