We study the Temporal Exploration problem, where an agent must visit all vertices of a temporal graph while traversing at most one available edge per time step. Unlike static graphs, which can be explored in linear time, temporal constraints can substantially increase exploration time even when every snapshot of the graph is connected. To better understand the source of this complexity, we focus on a near-static setting and consider always-connected $k$-edge-deficient temporal graphs, in which each snapshot is connected and differs from a fixed underlying $n$-vertex graph by at most $k$ edges. Although such graphs are structurally close to static graphs, they can still exhibit non-trivial temporal behaviour. Prior work showed that these graphs can be explored in $O(kn \log n)$ time steps and established a lower bound of $Ω(n \log k)$, leaving open whether linear-time exploration in $n$ is possible. We resolve this question by showing that any always-connected $k$-edge-deficient temporal graph admits an exploration schedule of length $O(nk \log k)$. Moreover, given such a temporal graph, the corresponding exploration schedule can be computed in polynomial time. The obtained bound is linear in the number of vertices up to a factor depending only on $k$, removes the extraneous logarithmic dependence on $n$, and is nearly optimal. In particular, for constant $k$, our result yields an order-optimal $Θ(n)$ exploration time, showing that temporal exploration in this near-static regime essentially retains the linear-time character of static graph traversal.

翻译：我们研究时间探索问题，其中智能体必须在每个时间步最多遍历一条可用边的条件下访问时间图的所有顶点。与可在线性时间内探索的静态图不同，即使图的每个快照都是连通的，时间约束也可能显著增加探索时间。为了更深入理解这种复杂性的来源，我们聚焦于近静态场景，并考虑始终连通的$k$-边缺陷时间图，其中每个快照都是连通的，且与一个固定的底层$n$顶点图相差至多$k$条边。尽管这类图在结构上接近静态图，但它们仍可能表现出非平凡的时间行为。先前研究表明，这些图可以在$O(kn \log n)$个时间步内探索，并建立了$Ω(n \log k)$的下界，从而留出了是否可以在$n$的线性时间内实现探索的开放问题。我们通过证明任意始终连通的$k$-边缺陷时间图都存在长度为$O(nk \log k)$的探索调度来解决该问题。此外，给定这样的时间图，相应的探索调度可以在多项式时间内计算得出。所得边界在顶点数上呈线性，仅依赖于$k$的因子，去除了对$n$多余的依赖，且近乎最优。特别地，对于常数$k$，我们的结果给出了阶最优的$Θ(n)$探索时间，表明该近静态场景下的时间探索本质上保留了静态图遍历的线性时间特性。