We study deterministic exploration by a single agent in $T$-interval-connected graphs, a standard model of dynamic networks in which, for every time window of length $T$, the intersection of the graphs within the window is connected. The agent does not know the window size $T$, nor the number of nodes $n$ or edges $m$, and must visit all nodes of the graph. We consider two visibility models, $KT_0$ and $KT_1$, depending on whether the agent can observe the identifiers of neighboring nodes. We investigate two fundamental questions: the minimum window size that guarantees exploration, and the optimal exploration time under sufficiently large window size. For both models, we show that a window size $T = Ω(m)$ is necessary. We also present deterministic algorithms whose required window size is $O(ε(n,m)\cdot m + n \log^2 n)$, where $ε(n,m) = \frac{\ln n}{1 + \ln m - \ln n}$. These bounds are tight for a wide range of $m$, in particular when $m = n^{1+Θ(1)}$. The same algorithms also yield optimal or near-optimal exploration time: we prove lower bounds of $Ω((m - n + 1)n)$ in the $KT_0$ model and $Ω(m)$ in the $KT_1$ model, and show that our algorithms match these bounds up to a polylogarithmic factor, while being fully time-optimal when $m = n^{1+Θ(1)}$. This yields tight bounds when parameterized solely by $n$: $Θ(n^3)$ for $KT_0$ and $Θ(n^2)$ for $KT_1$.

翻译：我们研究单个智能体在$T$-区间连通图中的确定性探索问题。$T$-区间连通图是动态网络的标准模型，其中每个长度为$T$的时间窗口内，窗口内所有图的交集是连通的。智能体既不知道窗口大小$T$，也不知道节点数$n$或边数$m$，且必须访问图中所有节点。我们考虑两种可见性模型：$KT_0$和$KT_1$，其区别在于智能体能否观察到相邻节点的标识符。我们探究两个基本问题：保证探索所需的最小窗口大小，以及在足够大窗口下的最优探索时间。对于两种模型，我们证明窗口大小$T = Ω(m)$是必要的。同时，我们提出确定性算法，其所需窗口大小为$O(ε(n,m)\cdot m + n \log^2 n)$，其中$ε(n,m) = \frac{\ln n}{1 + \ln m - \ln n}$。对于广泛范围的$m$，特别是当$m = n^{1+Θ(1)}$时，这些界是紧的。相同的算法还实现了最优或接近最优的探索时间：我们在$KT_0$模型中证明了下界$Ω((m - n + 1)n)$，在$KT_1$模型中证明了下界$Ω(m)$，并表明我们的算法与这些界仅相差一个多对数因子，且在$m = n^{1+Θ(1)}$时达到完全时间最优。这给出了仅以$n$为参数时的紧界：$KT_0$模型为$Θ(n^3)$，$KT_1$模型为$Θ(n^2)$。