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Minimum+1 (s,t)-cuts and Dual Edge Sensitivity Oracle Baswana, Surender Bhanja, Koustav Pandey, Abhyuday mincut maxflow fault tolerant Let G be a directed multi-graph on n vertices and m edges with a designated source vertex s and a designated sink vertex t. We study the (s,t)-cuts of capacity minimum+1 and as an important application of them, we give a solution to the dual edge sensitivity for (s,t)-mincuts - reporting the (s,t)-mincut upon failure or addition of any pair of edges. Picard and Queyranne [Mathematical Programming Studies, 13(1):8-16, 1980] showed that there exists a directed acyclic graph (DAG) that compactly stores all minimum (s,t)-cuts of G. This structure also acts as an oracle for the single edge sensitivity of minimum (s,t)-cut. Dinitz and Nutov [STOC, pages 509-518, 1995] showed that there exists an 𝒪(n) size 2-level cactus model that stores all global cuts of capacity minimum+1. However, for minimum+1 (s,t)-cuts, no such compact structures exist till date. We present the following structural and algorithmic results on minimum+1 (s,t)-cuts. 1) There exists a pair of DAGs of size O(m) that compactly store all minimum+1 (s,t)-cuts of G. Each minimum+1 (s,t)-cut appears as a (s,t)-cut in one of the 2 DAGs and is 3-transversal - it intersects any path in the DAG at most thrice. 2) There exists an O(n²) size data structure that, given a pair of vertices {u,v} which are not separated by an (s,t)-mincut, can determine in 𝒪(1) time if there exists a minimum+1 (s,t)-cut, say (A,B), such that {s,u} ∈ A and {v,t} ∈ B; the corresponding cut can be reported in 𝒪(|B|) time. 3) There exists an O(n²) size data structure that solves the dual edge sensitivity problem for (s,t)-mincuts. It takes 𝒪(1) time to report the value of a resulting (s,t)-mincut (A,B) and 𝒪(|B|) time to report the cut. 4) For the data structure problems addressed in (2) and (3) above, we also provide a matching conditional lower bound. We establish a close relationship among three seemingly unrelated problems – all-pairs directed reachability problem, the dual edge sensitivity problem for (s,t)-mincuts, and 2× 2 maximum flow. Assuming the directed reachability hypothesis, this leads to Ω(n²) lower bounds on the space for the latter two problems. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Surender Baswana and Koustav Bhanja and Abhyuday Pandey 2022 Is Part Of LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2022.15 urn:nbn:de:0030-drops-163566 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.15 eng https://creativecommons.org/licenses/by/4.0/legalcode