Exploration of k-edge-deficient temporal graphs

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.