A temporal graph with lifetime L is a sequence of L graphs \(G_1, \ldots ,G_L\), called layers, all of which have the same vertex set V but can have different edge sets. The underlying graph is the graph with vertex set V that contains all the edges that appear in at least one layer. The temporal graph is always connected if each layer is a connected graph, and it is k-edge-deficient if each layer contains all except at most k edges of the underlying graph. For a given start vertex s, a temporal exploration is a temporal walk that starts at s, traverses at most one edge in each layer, and visits all vertices of the temporal graph. We show that always-connected, k-edge-deficient temporal graphs with sufficient lifetime can always be explored in \(O(kn \log n)\) time steps. We also construct always-connected, k-edge-deficient temporal graphs for which any exploration requires \(\varOmega (n \log k)\) time steps. For always-connected, 1-edge-deficient temporal graphs, we show that O(n) time steps suffice for temporal exploration.

具有生命周期L的时间图是L个图的序列\(G_1, \ldots ,G_L\)，称为层，所有层都具有相同的顶点集 V，但可以具有不同的边集。底层图是具有顶点集V的图，其中包含至少出现在一层中的所有边。如果每一层都是连通图，则时间图始终是连通的，并且如果每一层包含除了底层图的最多k个边之外的所有边，则它是k边缺陷的。对于给定的起始顶点 s，时间探索是从 s开始的时间游走，在每一层中最多遍历一条边，并访问时间图的所有顶点。我们表明，始终可以在\(O(kn \log n)\)时间步中探索具有足够生命周期的始终连接的、缺乏k边的时间图。我们还构建了始终连接的、k边不足的时间图，任何探索都需要\(\varOmega (n \log k)\) 个时间步长。对于始终连接的、缺少 1 条边的时间图，我们证明O ( n ) 时间步足以进行时间探索。