We study the problem of finding a family of diverse minimum edge s-t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s-t cuts C_1, …, C_k such that for any i,j ∈ [k], the number of edges in the symmetric difference C_i △ C_j is at least d. For d ∈ {1,2}, the problem corresponds to counting minimum s-t cuts in G, which is #P-complete [Provan and Ball, SICOMP 1983]. The problem is also known to be NP-complete already for k = 3 [de Berg, López Martínez, Spieksma, ISAAC 2024]. Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d. The main ingredients of our FPT algorithm build on novel structural properties of diverse minimum s-t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlström [JACM 2025].

We study the problem of connectivity augmentation of a planar graph, while preserving planarity. This problem is motivated by many real-world settings such as road-networks, power-networks etc. In these settings, it is crucial to preserve the original planar embedding after augmentation. In 2009, Gutwenger and Mutzel gave a constructive algorithm showing that a connected planar graph with a fixed embedding (a plane graph) can be optimally augmented to a biconnected graph without crossings while preserving the embedding. We further this line of research, by giving an algorithm that computes a minimum set of edges that makes a connected plane graph 2-edge-connected in O(|V|(1+α(|V|))) time and linear space, where α is the inverse Ackermann function. We also study the 3-vertex-connectivity augmentation of biconnected outerplanar plane graphs. We present the first polynomial-time algorithm that augments such graphs to 3-connectivity with the minimum number of edges in O(|V|(1+α(|V|))) time and linear space while preserving the embedding, i.e. the augmented graph has a planar embedding that extends the given embedding.