Let G be a directed graph on n vertices and m edges. In this article, we study (s,t)-cuts of second minimum capacity and present the following algorithmic and graph-theoretic results. 1) Second (s,t)-mincut: Vazirani and Yannakakis [ICALP 1992] designed the first algorithm for computing an (s,t)-cut of second minimum capacity using {O}(n²) maximum (s,t)-flow computations. We present the following algorithm that improves the running time significantly. For directed integer-weighted graphs, there is an algorithm that can compute an (s,t)-cut of second minimum capacity using Õ(√n) maximum (s,t)-flow computations with high probability. To achieve this result, a close relationship of independent interest is established between (s,t)-cuts of second minimum capacity and global mincuts in directed weighted graphs. 2) Minimum+1 (s,t)-cuts: Minimum+1 (s,t)-cuts have been studied quite well recently [Baswana, Bhanja, and Pandey, ICALP 2022 & TALG 2023], which is a special case of second (s,t)-mincut. We present the following structural result and the first nontrivial algorithm for minimum+1 (s,t)-cuts. 3) Algorithm: For directed multi-graphs, we design an algorithm that, given any maximum (s,t)-flow, computes a minimum+1 (s,t)-cut, if it exists, in O(m) time. 4) Structure: The existing structures for storing and characterizing all minimum+1 (s,t)-cuts occupy {O}(mn) space [Baswana, Bhanja, and Pandey, TALG 2023]. For undirected multi-graphs, we design a directed acyclic graph (DAG) occupying only {O}(m) space that stores and characterizes all minimum+1 (s,t)-cuts. This matches the space bound of the widely-known DAG structure for all (s,t)-mincuts [Picard and Queyranne, Math. Prog. Studies 1980]. 5) Dual Edge Sensitivity Oracle: The study of minimum+1 (s,t)-cuts often turns out to be useful in designing dual edge sensitivity oracles - a compact data structure for efficiently reporting an (s,t)-mincut after insertion/failure of any given pair of query edges. It has been shown recently [Bhanja, ICALP 2025] that any dual edge sensitivity oracle for (s,t)-mincut in undirected multi-graphs must occupy Ω(n²) space in the worst-case irrespective of the query time. Interestingly, for undirected unweighted simple graphs, we break this quadratic barrier while achieving a non-trivial query time as follows. There is an O(n√n) space data structure that can report an (s,t)-mincut in O(min{m,n√n}) time after the insertion/failure of any given pair of query edges. To arrive at our results, as one of our key techniques, we establish interesting relationships between (s,t)-cuts of capacity (minimum+Δ), Δ ≥ 0, and maximum (s,t)-flow. We believe that these techniques and the graph-theoretic result in 2.(b) are of independent interest.

We present a compact labeling scheme for determining whether a designated set of terminals in a graph remains connected after any f (or less) vertex failures occur. An f-FT Steiner connectivity labeling scheme for an n-vertex graph G = (V,E) with terminal set U ⊆ V provides labels to the vertices of G, such that given only the labels of any subset F ⊆ V with |F| ≤ f, one can determine if U remains connected in G-F. The main complexity measure is the maximum label length. The special case U = V of global connectivity has been recently studied by Jiang, Parter, and Petruschka [Yonggang Jiang et al., 2025], who provided labels of n^{1-1/f} ⋅ poly(f,log n) bits. This is near-optimal (up to poly(f,log n) factors) by a lower bound of Long, Pettie and Saranurak [Yaowei Long et al., 2025]. Our scheme achieves labels of |U|^{1-1/f} ⋅ poly(f, log n) for general U ⊆ V, which is near-optimal for any given size |U| of the terminal set. To handle terminal sets, our approach differs from [Yonggang Jiang et al., 2025]. We use a well-structured Steiner tree for U produced by a decomposition theorem of Duan and Pettie [Ran Duan and Seth Pettie, 2020], and bypass the need for Nagamochi-Ibaraki sparsification [Hiroshi Nagamochi and Toshihide Ibaraki, 1992].

Let $G=(V,E)$ be an undirected multi-graph on $n=|V|$ vertices and $S\subseteq V$ be a Steiner set. Steiner cut is a fundamental concept; moreover, global cut $(|S|=n)$, as well as (s,t)-cut $(|S|=2)$, is just a special case of Steiner cut. We study Steiner cuts of capacity minimum+1, and as an important application, we provide a dual edge Sensitivity Oracle for Steiner mincut. A compact data structure for cuts of capacity minimum+1 has been designed for both global cuts [STOC 1995] and (s,t)-cuts [TALG 2023]. Moreover, both data structures are also used crucially to design a dual edge Sensitivity Oracle for their respective mincuts. Unfortunately, except for these two extreme scenarios of Steiner cuts, no generalization of these results is known. Therefore, to address this gap, we present the following first results on Steiner cuts. 1. Data Structure: There is an $O(n(n-|S|+1))$ space data structure that can determine in $O(1)$ time whether a given pair of vertices is separated by a Steiner cut of capacity at least minimum+1. It can report such a cut, if it exists, in $O(n)$ time. 2. Sensitivity Oracle: (a) There is an $O(n(n-|S|+1))$ space data structure that, after the failure/insertion of any pair of edges, can report the capacity of Steiner mincut in $O(1)$ time and a Steiner mincut in $O(n)$ time. (b) If we are interested in reporting only the capacity, there is a more compact data structure that occupies $O((n-|S|)^2+n)$ space and reports the capacity in $O(1)$ time after the failure/insertion of any pair of edges. 3. Lower Bound: For undirected multi-graphs, for every Steiner set $S$, any data structure that, after the failure or insertion of any pair of edges, can report the capacity of Steiner mincut must occupy $Ω((n-|S|)^2)$ bits of space, irrespective of the query time.

Let G be a directed weighted graph (DiGraph) on n vertices and m edges with source s and sink t. An edge in G is vital if its removal reduces the capacity of (s,t)-mincut. Since the seminal work of Ford and Fulkerson, a long line of work has been done on computing the most vital edge and all vital edges of G. Unfortunately, after 60 years, the existing results are for undirected or unweighted graphs. We present the following result for DiGraph, which solves an open problem stated by Ausiello et al. 1. There is an algorithm that computes all vital edges as well as the most vital edge of G using O(n) maxflow computations. Vital edges play a crucial role in the design of Sensitivity Oracle (SO) for (s,t)-mincut. For directed graphs, the only existing SO is for unweighted graphs by Picard and Queyranne. We present the first and optimal SO for DiGraph. 2. (a) There is an O(n) space SO that can report in O(1) time the capacity of (s,t)-mincut and (b) an O($n^2$) space SO that can report an (s,t)-mincut in O(n) time after failure/insertion of an edge. For unweighted graphs, Picard and Queyranne designed an O(m) space DAG that stores and characterizes all mincuts for all vital edges. Conversely, there is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge belongs to the set. We generalize these results for DiGraph. 3. (a) There is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge is present in the set. (b) We design two compact structures for storing and characterizing all mincuts for all vital edges, (i) O(m) space DAG for partial characterization and (ii) O(mn) space structure for complete characterization. To arrive at our results, we develop new techniques, especially a generalization of maxflow-mincut theorem by Ford and Fulkerson, which might be of independent interest.

Let $G=(V,E)$ be an undirected weighted graph on $n=|V|$ vertices and $S\subseteq V$ be a Steiner set. Steiner mincut is a well-studied concept, which provides a generalization to both (s,t)-mincut (when $|S|=2$) and global mincut (when $|S|=n$). Here, we address the problem of designing a compact data structure that can efficiently report a Steiner mincut and its capacity after the failure of any edge in $G$; such a data structure is known as a \textit{Sensitivity Oracle} for Steiner mincut. In the area of minimum cuts, although many Sensitivity Oracles have been designed in unweighted graphs, however, in weighted graphs, Sensitivity Oracles exist only for (s,t)-mincut [Annals of Operations Research 1991, NETWORKS 2019, ICALP 2024], which is just a special case of Steiner mincut. Here, we generalize this result to any arbitrary set $S\subseteq V$. 1. Sensitivity Oracle: Assuming the capacity of every edge is known, a. there is an ${\mathcal O}(n)$ space data structure that can report the capacity of Steiner mincut in ${\mathcal O}(1)$ time and b. there is an ${\mathcal O}(n(n-|S|+1))$ space data structure that can report a Steiner mincut in ${\mathcal O}(n)$ time after the failure of any edge in $G$. 2. Lower Bound: We show that any data structure that, after the failure of any edge, can report a Steiner mincut or its capacity must occupy $Ω(n^2)$ bits of space in the worst case, irrespective of the size of the Steiner set. The lower bound in (2) shows that the assumption in (1) is essential to break the $Ω(n^2)$ lower bound on space. For $|S|=n-k$ for any constant $k\ge 0$, it occupies only ${\mathcal O}(n)$ space. So, we also present the first Sensitivity Oracle occupying ${\mathcal O}(n)$ space for global mincut.

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.