A temporal graph is a graph for which the edge set can change from one time step to the next. This paper considers undirected temporal graphs defined over L time steps and connected at each time step. We study the Shortest Temporal Exploration Problem (STEXP) that, given the evolution of the graph, asks for a temporal walk that starts at a given vertex, moves over at most one edge at each time step, visits all the vertices, takes at most L time steps and traverses the smallest number of edges. We prove that every constantly connected temporal graph with n vertices can be explored with O(n^{1.5}) edges traversed if L is O(n^{3.5}) time steps. This result improves the upper bound of O(n²) edges when L is Ω(n²). Moreover, we study the case where the graph has a diameter bounded by a parameter k at each time step and we prove that there exists an exploration which takes O(kn²) time steps and traverses O(kn) edges. Finally, the case where the underlying graph is a cycle is studied and tight linear bounds are provided on the number of edges traversed in the worst-case.
@InProceedings{balev_et_al:LIPIcs.SAND.2025.18, author = {Balev, Stefan and Sanlaville, \'{E}ric and Toullalan, Antoine}, title = {{Brief Announcement: The Shortest Temporal Exploration Problem}}, booktitle = {4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)}, pages = {18:1--18:5}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-368-3}, ISSN = {1868-8969}, year = {2025}, volume = {330}, editor = {Meeks, Kitty and Scheideler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.18}, URN = {urn:nbn:de:0030-drops-230716}, doi = {10.4230/LIPIcs.SAND.2025.18}, annote = {Keywords: Graph Theory, Temporal Graph, Temporal Graph Exploration} }
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