A spanner of a temporal graph is a subset of edges that preserves connectivity over time between vertices. A minimal spanner is one in which no additional edges can be removed without breaking this connectivity. Our focus is on enumerating minimal spanners for a given temporal graph. We explore several variations of this problem based on the type of connectivity that must be maintained, ranging from one-to-all connectivity to one-to-all-to-one, many-to-all, and finally all-to-all connectivity. We establish that these problems become progressively harder: (i) We present a polynomial-delay enumeration algorithm for one-to-all connectivity; (ii) We prove Dual-hardness for both one-to-all-to-one and many-to-all connectivity, even in the restricted case of two-to-all; (iii) Finally, for all-to-all connectivity, we show that enumeration cannot be performed in output-polynomial time unless P = NP.
@InProceedings{kurita_et_al:LIPIcs.SAND.2025.9, author = {Kurita, Kazuhiro and Marino, Andrea and Schoeters, Jason and Uno, Takeaki}, title = {{Spanner Enumeration for Temporal Graphs}}, booktitle = {4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)}, pages = {9:1--9:21}, 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.2025.9}, URN = {urn:nbn:de:0030-drops-230621}, doi = {10.4230/LIPIcs.SAND.2025.9}, annote = {Keywords: temporal graphs, temporal spanners, one-to-all connectivity, all-to-all connectivity enumeration, NP-completeness, Dual-hardness, binary partition tree, flashlight search, polynomial delay} }
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