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 introduce a series of graph decompositions based on the modulator/target scheme of modification problems that enable several algorithmic applications that parametrically extend the algorithmic potential of planarity. In the core of our approach is a polynomial time algorithm for computing planar \(\mathcal{H}\)-modulators. Given a graph class \(\mathcal{H}\), a planar \(\mathcal{H}\)-modulator of a graph \(G\) is a set \(X \subseteq V(G)\) such that the “torso” of \(X\) is planar and all connected components of \(G-X\) belong to \(\mathcal{H}\). Here, the torso of \(X\) is obtained from \(G[X]\) if, for every connected component of \(G-X\), we form a clique out of its neighborhood on \(G[X]\). We introduce \(\mathcal{H}\)-Planarity as the problem of deciding whether a graph \(G\) has a planar \(\mathcal{H}\)-modulator. We prove that, if \(\mathcal{H}\) is hereditary, CMS0-definable, and decidable in polynomial time, then \(\mathcal{H}\)-Planarity is solvable in polynomial time.

In the problem Fault-Tolerant Path ( FTP ), we are given an edge-weighted directed graph G = ( V , E ) , a subset U ⊆ E of vulnerable edges, two vertices s, t ∈ V , and integers k and ℓ. The task is to decide whether there exists a subgraph H of G with total cost at most ℓ such that, after the removal of any k vulnerable edges, H still contains an s − t -path. We study whether Fault-Tolerant Path is fixed-parameter tractable (FPT) and whether it admits a polynomial kernel under various parameterizations. Our choices of parameters include: the number of vulnerable edges in the input graph, the number of safe (i.e, invulnerable) edges in the input graph, the budget ℓ, the minimum number of safe edges in any optimal solution, the minimum number of vulnerable edges in any optimal solution, the required redundancy k , and natural above- and below-guarantee parameterizations. We provide an almost complete description of the complexity landscape of FTP for these parameters.

We investigate the problem of constructing fault-tolerant bases in matroids. Given a matroid M and a redundancy parameter k, a k-fault-tolerant basis is a minimum-size set of elements such that, even after the removal of any k elements, the remaining subset still spans the entire ground set. Since matroids generalize linear independence across structures such as vector spaces, graphs, and set systems, this problem unifies and extends several fault-tolerant concepts appearing in prior research. Our main contribution is a fixed-parameter tractable (FPT) algorithm for the k-fault-tolerant basis problem, parameterized by both k and the rank r of the matroid. This two-variable parameterization by k + r is shown to be tight in the following sense. On the one hand, the problem is already NP-hard for k=1. On the other hand, it is Para-NP-hard for r \geq 3 and polynomial-time solvable for r \leq 2.

A replacement action is a function ℒ that maps each graph H to a collection of graphs of size at most |V(H)|. Given a graph class ℋ, we consider a general family of graph modification problems, called ℒ-Replacement to ℋ, where the input is a graph G and the question is whether it is possible to replace some induced subgraph H₁ of G on at most k vertices by a graph H₂ in ℒ(H₁) so that the resulting graph belongs to ℋ. ℒ-Replacement to ℋ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves ℒ-Replacement to ℋ in time 2^poly(k) ⋅ |V(G)|² for every minor-closed graph class ℋ, where poly is a polynomial whose degree depends on ℋ, under a mild technical condition on ℒ. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to ℋ within the same running time. Our second algorithm is an improvement of the first one when ℋ is the class of graphs embeddable in a surface of Euler genus at most g and runs in time 2^𝒪(k⁹) ⋅ |V(G)|², where the 𝒪(⋅) notation depends on g. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Let G be a minor-closed graph class and let G be an n-vertex graph. We say that G is a k-apex of G if G contains a set S of at most k vertices such that G⧵S belongs to G. Our first result is an algorithm that decides whether G is a k-apex of G in time 2^poly(k)⋅n². This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020, TALG 2022], whose running time was 2^poly(k)⋅n³. The elimination distance of G to G, denoted by ed_G(G), is the minimum number of rounds required to reduce each connected component of G to a graph in G by removing one vertex from each connected component in each round. Bulian and Dawar [Algorithmica 2017] proved the existence of an FPT-algorithm, with parameter k, to decide whether ed_G(G) ≤ k. This algorithm is based on the computability of the minor-obstructions and its dependence on k is not explicit. We extend the techniques used in the first algorithm to decide whether ed_G(G) ≤ k in time 2^{2^{2^poly(k)}}⋅n². This is the first algorithm for this problem with an explicit parametric dependence in k. In the special case where G excludes some apex-graph as a minor, we give two alternative algorithms, one running in time 2^{2^O(k²log k)}⋅n² and one running in time 2^{poly(k)}⋅n³. As a stepping stone for these algorithms, we provide an algorithm that decides whether ed_G(G) ≤ k in time 2^O(tw⋅ k + tw log tw)⋅n, where tw is the treewidth of G. This algorithm combines the dynamic programming framework of Reidl, Rossmanith, Villaamil, and Sikdar [ICALP 2014] for the particular case where G contains only the empty graph (i.e., for treedepth) with the representative-based techniques introduced by Baste, Sau, and Thilikos [SODA 2020]. In all the algorithmic complexities above, poly is a polynomial function whose degree depends on G, while the hidden constants also depend on G. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs E_k(G) = {G ∣ ed_G(G) ≤ k}.

We describe Touiouidth, a twin-width solver for the exact-track of the 2023 PACE Challenge: Twin Width. Our solver is based on a simple branch and bound algorithm with search space reductions and is implemented in C++.

We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Theor. Comput. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that K_t-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are FPT parameterized by bipartite treewidth. We also provide the following complexity dichotomy when H is a 2-connected graph, for each of the H-Subgraph-Packing, H-Induced-Packing, H-Scattered-Packing, and H-Odd-Minor-Packing problems: if H is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if H is non-bipartite, then the problem is solvable in XP-time. Beyond bipartite treewidth, we define 1-ℋ-treewidth by replacing the bipartite graph class by any graph class ℋ. Most of the technology developed here also works for this more general parameter.