Exploration of k-edge-deficient temporal graphs

Acta Informatica (2022) 59:387–407 https://doi.org/10.1007/s00236-022-00421-5 ORIGINAL ARTICLE Exploration of k-edge-deficient temporal graphs Thomas Erlebach1 · Jakob T. Spooner2 Received: 10 September 2021 / Accepted: 28 March 2022 / Published online: 26 August 2022 © The Author(s) 2022 Abstract A temporal graph with lifetime L is a sequence of L graphs G 1 , . . . , 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-edgedeficient 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 Ω(n log k) time steps. For always-connected, 1-edge-deficient temporal graphs, we show that O(n) time steps suffice for temporal exploration. 1 Introduction Given a simple, connected, undirected graph G and a start vertex s ∈ V (G), the task of exploring G, i.e. computing a sequence of consecutively crossed edges e ∈ E(G) that begins at s and visits every vertex v ∈ V (G) at least once, is both natural and well-understood. A closely related problem was initially considered by Shannon [23], who designed a mechanical maze-solving machine which implemented a depth-first search-type technique in order to locate, within a given maze, a prespecified goal. This ‘searching’ problem is indeed related to graph exploration: if our task is to simply complete an exploration of G, then a solution can be straightforwardly found by performing a depth-first search (DFS) starting from s and An extended abstract of a preliminary version of this paper has been presented at the 17th Algorithms and Data Structures Symposium (WADS 2021) [10]. T. Erlebach’s research was supported by EPSRC Grants EP/S033483/1 and EP/T01461X/1. B Thomas Erlebach Jakob T. Spooner 1 Department of Computer Science, Durham University, Durham, England 2 School of Computing and Mathematical Sciences, University of Leicester, Leicester, England 123 388 T. Erlebach, J. T. Spooner stopping once all vertices have been visited at least once—clearly this requires Θ(n) edge traversals in total, where n = |V (G)|. The graph exploration problem in the context of temporal graphs (i.e. graphs whose edge set can change over time) has also received significant attention in recent years [1, 2, 6–9, 21]. This problem, known as Temporal Exploration (TEXP), but restricted to k-edgedeficient temporal graphs (which we define formally later) is the focus of this paper. Given a temporal graph G , the problem asks that we compute a temporal walk, starting at some prespecified vertex s ∈ V (G ), that makes at most a single edge traversal in each time step, and that visits all vertices at least once by the earliest time possible. We formally define the problem and temporal graph model in Sect. 2, but refer the interested reader to [5, 20] for more on temporal graphs in general, or [6, 21] for more details on TEXP. In a rather general setting, TEXP makes no assumptions about the input temporal graph, aside from the assumption that it is connected in each time step (i.e. always-connected), which ensures exploration is always possible provided the temporal graph has a sufficient lifetime [21]. This general setting allows an arbitrary number of edges from the underlying graph to be missing in each time step, and thus the graphs in different time steps can differ substantially, which leads to pessimistic bounds on the worst-case exploration time: it was shown by Erlebach et al. [6] that there are always-connected temporal graphs with n vertices that require Θ(n 2 ) time steps to be explored. The construction of temporal graphs with such a large exploration time from [6] uses an underlying graph that is dense (it has Θ(n 2 ) edges), whilst the graph in each time step is a star (with n − 1 edges). Thus, in each time step a quadratic number of edges of the underlying graph are missing from the graph of that time step. Therefore, it is interesting to study the question whether better exploration times can be guaranteed if the number of missing edges in each time step is small. To study this question, we also consider always-connected temporal graphs but, in contrast to previous work, we consider k-edgedeficient temporal graphs whose structure in each step is ‘close’ to that of its underlying graph, in the sense that at most k edges are missing. Such graphs were previously considered by Gotoh et al. [14] in a distributed setting. We assume that the temporal structure of an input temporal graph is known in full to an algorithm prior to it computing a solution, as opposed to a setting in which the structure of the graph in each step is revealed online and over time. 1.1 Our contribution We consider Temporal Exploration on always-connected temporal graphs that are kedge-deficient for some k ∈ N. We define the property formally in Sect. 2, but essentially these are temporal graphs G with underlying graph G such that, during each time step t of G ’s lifetime, there are at most k edges e ∈ E in the underlying graph that are untraversable in (or ‘missing’ from) G . Let n = |V (G)|. In Sect. 3, we prove for arbitrary k ∈ N that kedge-deficient always-connected temporal graphs can be explored in O(kn log n) time steps. In Sect. 4, we additionally show that 1-edge-deficient graphs can always be explored in 51n time steps, by giving a recursive exploration algorithm that exploits a number of existing structural/algorithmic results originating from traditional graph theory. In Sect. 5, we present a modification of an existing Ω(n log n) lower bound on the number of time steps required to explore always-connected temporal graphs with planar underlying graph of maximum degree ≤ 4, presented in [6], that allows us to obtain an Ω(n log k) bound on the worst-case time required to explore arbitrary always-connected k-edge-deficient temporal graphs. Finally, we conclude and point to directions for future work in Sect. 6. 123 Exploration of k-edge-deficient temporal graphs 389 1.2 Related work Bui-Xuan et al. [4] propose multiple objectives for optimization when computing temporal walks/paths from one vertex to another: e.g. fastest (fewest steps used) and foremost (arriving at the destination at the earliest time possible). Brodén et al. [3] consider the Temporal Travelling Salesperson Problem on a complete graph with n vertices, with edge costs that can differ between 1 and 2 in each tim (...truncated)