发表机构
University of Geneva（日内瓦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对度有界的时间图，构造确定性下界实例，证明探索时间至少为 $\gamma\Delta n(1+\log(n/\Delta))$，利用旋转循环和格雷码重分配隐藏顶点。
AI 中文摘要
时间图是一组图在公共顶点集 $n$ 上的序列，即快照，每个时间步对应一个快照。一个智能体，预先知道整个序列，可以在每个时间步等待或沿着当前快照的一条边移动，当智能体访问了所有顶点时，时间图即被探索。我们考虑始终连通的时间图，其中每个快照都是连通的，并询问当底层图（所有快照的并集）的最大度至多为 $\Delta$ 时，探索可以被强制持续多长时间。在此设置下，已知两个下界：$\Omega(\Delta n)$ 和 $\Omega(n\log n)$，由不同的构造实现。我们建立了一个更强的下界，回答了 Bastide、Groenland、Michel 和 Rambaud 的一个问题。具体来说，对于每个 $n$ 和 $\Delta$，其中 $\Delta_0\le\Delta\le n-1$，我们构造一个在 $n$ 个顶点上的始终连通时间图，其底层最大度至多 $\Delta$，从任何起始顶点出发，探索所需的时间步数不少于 $\gamma\\,\Delta n\\,(1+\log(n/\Delta))$，其中 $\gamma>0$ 和 $\Delta_0$ 是绝对常数。该构造是确定性的，并且每个快照是一棵生成树，其中恰好有一个顶点的度大于三。时间被划分为多个阶段。在每个阶段中，一个旋转循环装置阻止智能体到达其附着的树中的超过一半。在阶段之间，我们使用常数度扩展图上的行走在这些树之间重新分配目标顶点，并且目标的格雷码顺序保持底层度有界。重新分配保证了对于智能体的任何行走，某个目标在所有阶段中始终保持未被访问。
英文摘要
A temporal graph is a sequence of graphs on a common set of $n$ vertices, its snapshots, one for each time step. An agent, knowing the entire sequence in advance, may at each time step wait or move along an edge of the current snapshot, and the temporal graph is explored once the agent has visited every vertex. We consider always-connected temporal graphs, in which every snapshot is connected, and ask how long exploration can be forced to take when the underlying graph, the union of all snapshots, has maximum degree at most~$Δ$. Two lower bounds were known in this setting, $Ω(Δn)$ and $Ω(n\log n)$, realised by different constructions. We establish a stronger lower bound, answering a question of Bastide, Groenland, Michel and Rambaud. Specifically, for every $n$ and $Δ$ with $Δ_0\leΔ\le n-1$ we construct an always-connected temporal graph on $n$ vertices with underlying maximum degree at most~$Δ$ that cannot be explored in fewer than $γ\,Δn\,(1+\log(n/Δ))$ time steps from any start vertex, where $γ>0$ and $Δ_0$ are absolute constants. The construction is deterministic, and every snapshot is a spanning tree with exactly one vertex of degree greater than three. Time is divided into phases. In each phase, a rotating-cycle gadget prevents the agent from reaching more than half of the trees attached to it. Between phases, we reassign target vertices among these trees using walks on a constant-degree expander, and a Gray code order of the targets keeps the underlying degree bounded. The reassignment guarantees that, for any walk of the agent, some target remains unvisited throughout all phases.
Comments21 pages, 2 figures
