\emph{Temporal graphs} are a generalisation of (static) graphs, defined by a sequence of \emph{snapshots}, each a static graph defined over a common set of vertices. \emph{Exploration} problems are one of the most fundamental and most heavily studied problems on temporal graphs, asking if a set of $m$ agents can visit every vertex in the graph, with each agent only allowed to traverse a single edge per snapshot. In this paper, we introduce and study \emph{always $S$-connected} temporal graphs, a generalisation of always connected temporal graphs where, rather than forming a single connected component in each snapshot, we have at most $\vert S \vert$ components, each defined by the connection to a single vertex in the set $S$. We use this formulation as a tool for exploring graphs admitting an \emph{$(r,b)$-division}, a partitioning of the vertex set into disconnected components, each of which is $S$-connected, where $\vert S \vert \leq b$. We show that an always $S$-connected temporal graph with $m = \vert S \vert$ and an average degree of $Δ$ can be explored by $m$ agents in $O(n^{1.5} m^3 Δ^{1.5}\log^{1.5}(n))$ snapshots. Using this as a subroutine, we show that any always-connected temporal graph with treewidth at most $k$ can be explored by a single agent in $O\left(n^{4/3} k^{5.5}\log^{2.5}(n)\right)$ snapshots, improving on the current state-of-the-art for small values of $k$. Further, we show that interval graph with only a small number of large cliques can be explored by a single agent in $O\left(n^{4/3} \log^{2.5}(n)\right)$ snapshots.

翻译：\emph{时序图}是（静态）图的推广，由一系列\emph{快照}定义，每个快照是在公共顶点集上定义的静态图。\emph{探索}问题是时序图上最基本且研究最深入的问题之一，询问一组$m$个智能体能否访问图中的每个顶点，其中每个智能体在每个快照中只允许遍历一条边。本文引入并研究\emph{始终$S$连通}时序图，这是始终连通时序图的推广。在始终$S$连通时序图中，每个快照最多形成$\vert S \vert$个连通分量，每个分量通过与集合$S$中的单个顶点连接来定义。我们利用这一公式作为探索具有\emph{$(r,b)$-划分}图的工具，该划分将顶点集分割为互不连通的分量，每个分量都是$S$连通的，其中$\vert S \vert \leq b$。我们证明，一个具有$m = \vert S \vert$且平均度为$Δ$的始终$S$连通时序图，可由$m$个智能体在$O(n^{1.5} m^3 Δ^{1.5}\log^{1.5}(n))$个快照内完成探索。以此作为子程序，我们证明任何树宽至多为$k$的始终连通时序图，可由单个智能体在$O\left(n^{4/3} k^{5.5}\log^{2.5}(n)\right)$个快照内完成探索，这改进了当前在$k$值较小时的最佳结果。此外，我们证明仅包含少量大团的区间图，可由单个智能体在$O\left(n^{4/3} \log^{2.5}(n)\right)$个快照内完成探索。