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 an ( s,t )-mincut upon failure or insertion 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. For undirected multi-graphs, Dinitz and Nutov [STOC, 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 structure exists till date. We present the following structural and algorithmic results on minimum+1 ( s,t )-cuts. (1) Structure: There is an 𝒪( m ) size 2-level DAG structure that stores all minimum+1 (s,t) -cuts of G such that each minimum+1 ( s,t )-cut appears as 3-transversal cut—it intersects any path in this structure at most thrice. We also show that there is an 𝒪( mn ) size structure for storing and characterizing all minimum+1 (s,t) -cuts in terms of 1-transversal cuts. (2) Data structure: There exists an 𝒪( n 2 ) size data structure that, given a pair of vertices {u,v} that 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) Sensitivity oracle: There exists an 𝒪( n 2 ) size data structure that solves the dual-edge sensitivity problem for (s,t) -mincuts. It takes 𝒪(1) time to report the capacity of a resulting (s,t) -mincut (A,B) and 𝒪(| B |) time to report the cut. (4) Lower bounds: For the data structure problems addressed in results (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 the problem of reporting the capacity of ({ x,y }, { u,v })-mincut for any four vertices x,y,u,v in G . Assuming the Directed Reachability Hypothesis by Patrascu [SIAM J. Computing, 827–847 (2011)] and Goldstein et al. [WADS, 421–436 (2017)], this leads to \(\tilde{\Omega }(n^2)\) lower bounds on the space for the latter two problems.