Non-strict Temporal Exploration

Erlebach, Thomas; Spooner, Jakob T.

doi:10.1007/978-3-030-54921-3_8

Cited by 12 publications

(11 citation statements)

References 20 publications

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“…We mention that, unlike the problems investigated here, a lot of research has been devoted to temporal node exploration, e.g. bounding the arrival time of such walks in special instances [9,8] and extending previous results in the case of non-strictly increasing paths [10].…”

Section: Introductionmentioning

confidence: 56%

Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

Marino¹,

Silva²

2021

Preprint

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An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. What if Euler had to take a bus? In a temporal graph (G, λ), with λ : E(G) → 2 [τ ] , an edge e ∈ E(G) is available only at the times specified by λ(e) ⊆ [τ ], in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this scenario, even though several translations of Eulerian trails and walks are possible in temporal terms, only a very particular variation has been exploited in the literature, specifically for infinite dynamic networks (Orlin, 1984). In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether (G, λ) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP-complete even if it has lifetime 2. In contrast, in the general case, solving any of these problems is NP-complete, even under very strict hypothesis.

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Section: Introductionmentioning

confidence: 56%

Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

Marino¹,

Silva²

2021

Preprint

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“…Generally, connectivity related problems have received a lot of attention in the temporal setting, ranging from the mentioned temporal path and betweenness computation to finding temporally connected subgraphs [6,20], temporal separation [35,46,50,54,68], temporal graph modification to decrease or increase its connectivity [24,27,28,55], temporal graph exploration [2,13,17,30,31,32], temporal network design [1,51], and others [38,40,48].…”

Section: Related Workmentioning

confidence: 99%

Counting Temporal Paths

Enright¹,

Meeks²,

Molter³

2022

Preprint

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The betweenness centrality of a vertex v is an important centrality measure that quantifies how many optimal paths between pairs of other vertices visit v. Computing betweenness centrality in a temporal graph, in which the edge set may change over discrete timesteps, requires us to count temporal paths that are optimal with respect to some criterion. For several natural notions of optimality, including foremost or fastest temporal paths, this counting problem reduces to #Temporal Path, the problem of counting all temporal paths between a fixed pair of vertices; like the problems of counting foremost and fastest temporal paths, #Temporal Path is #P-hard in general. Motivated by the many applications of this intractable problem, we initiate a systematic study of the prameterised and approximation complexity of #Temporal Path. We show that the problem presumably does not admit an FPT-algorithm for the feedback vertex number of the static underlying graph, and that it is hard to approximate in general. On the positive side, we proved several exact and approximate FPT-algorithms for special cases.

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“…Erlebach et al [7] prove an O(dn 1.75 ) bound on the number of time steps required to explore any temporal graph with degree bounded by d in each step, a considerable improvement over the previously best known O( n 2 log d log n ) bound [8]. In [9], a non-strict variant of TEXP is studied-here, a computed walk may make an unlimited number of edge traversals in each given time step. Notions of strict/non-strict paths which, respectively, allow for a single edge/unlimited number of edge(s) to be crossed in any time step have been considered before, notably by Kempe et al [19] and Zschoche et al [24].…”

Section: Related Workmentioning

confidence: 99%

“…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][7][8][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.…”

Section: Introductionmentioning

confidence: 99%

Exploration of k-edge-deficient temporal graphs

Erlebach

Spooner

2022

Acta Informatica

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A temporal graph with lifetime L is a sequence of L graphs $$G_1, \ldots ,G_L$$ 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-edge-deficient temporal graphs with sufficient lifetime can always be explored in $$O(kn \log n)$$ O ( k n log n ) time steps. We also construct always-connected, k-edge-deficient temporal graphs for which any exploration requires $$\varOmega (n \log k)$$ 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.

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Non-strict Temporal Exploration

Cited by 12 publications

References 20 publications

Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

Counting Temporal Paths

Exploration of k-edge-deficient temporal graphs

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