Computing edge-connected components in directed and undirected graphs is a fundamental and well-studied problem in graph algorithms. In a very recent breakthrough, Korhonen [STOC 2025] showed that for any fixed k, the k-edge connected components of an undirected graph can be computed in linear time. In contrast, the directed case remains significantly more challenging: linear-time algorithms are only known for k ≤ 3, and for any fixed k > 3, the best known bound for sparse or moderately dense graphs is still the O(mn)-time algorithm of Nagamochi and Watanabe (1993). In this paper, we break the O(mn) barrier for all k = o(n^{1/4}/√{log{n}}). We present a randomized algorithm that computes the (k+2)-edge-connected components of a k-edge-connected directed graph in O(k² m √n log n) time, for any k. This constitutes the first improvement over the classic Nagamochi-Watanabe bound for any constant k > 3. Our approach introduces new structural insights into directed edge-cuts and combines these with both new and existing techniques. A central contribution of our work is a substantial simplification and generalization of the framework introduced in [Loukas Georgiadis et al., 2023], which achieved an Õ(m√m) bound for computing the 3-edge-connected components of a digraph. In addition, we develop a variant of our algorithm that achieves the same O(m √n log n) running time for computing the 4-edge-connected components of a general directed graph.

Abstract Chordal graphs are characterized as the intersection graphs of subtrees in a tree and such a representation is known as the tree model. Restricting the characterization results in well-known subclasses of chordal graphs such as interval graphs or split graphs. A typical example of a problem that does not behave computationally the same in all subclasses of chordal graphs is the Subset Feedback Vertex Set (SFVS) problem: given a vertex-weighted graph $$G=(V,E)$$ G = ( V , E ) and a set $$S\subseteq V$$ S ⊆ V , we seek for a vertex set of minimum weight that intersects all cycles containing a vertex of S . SFVS is known to be polynomial-time solvable on interval graphs, whereas SFVS remains np -complete on split graphs and, consequently, on chordal graphs. Towards a better understanding of the complexity of SFVS on subclasses of chordal graphs, we exploit structural properties of a tree model in order to cope with the hardness of SFVS. Here we consider the leafage , which measures the minimum number of leaves in a tree model. We show that SFVS can be solved in polynomial time for every chordal graph with bounded leafage. In particular, given a chordal graph on n vertices with leafage $$\ell $$ ℓ , we provide an algorithm for solving SFVS with running time $$n^{O(\ell )}$$ n O ( ℓ ) , thus improving upon $$n^{O(\ell ^2)}$$ n O ( ℓ 2 ) , which is the running time of an approach that utilizes the previously known algorithm for graphs with bounded mim-width. We complement our result by showing that SFVS is w [1]-hard parameterized by $$\ell $$ ℓ . Pushing further our positive result, it is natural to also consider the vertex leafage , which measures the minimum upper bound on the number of leaves of every subtree in a tree model. However, we show that it is unlikely to obtain a similar result, as we prove that SFVS remains np -complete on undirected path graphs, i.e., chordal graphs having vertex leafage at most two. Lastly, we provide a polynomial-time algorithm for solving SFVS on rooted path graphs, a proper subclass of undirected path graphs and graphs with mim-width one, which is faster than the approach of constructing a graph decomposition of mim-width one and applying the previously known algorithm for graphs with bounded mim-width.

Abstract A simplicial vertex of a graph is a vertex whose neighborhood is a clique. It is known that listing all simplicial vertices can be done in O(nm) time or $$O(n^{\omega })$$ O ( n ω ) time, where $$O(n^{\omega })$$ O ( n ω ) is the time needed to perform a fast matrix multiplication. The notion of avoidable vertices generalizes the concept of simplicial vertices in the following way: a vertex u is avoidable if every induced path on three vertices with middle vertex u is contained in an induced cycle. We present algorithms for listing all avoidable vertices of a graph through the notion of minimal triangulations and common neighborhood detection. In particular we give algorithms with running times $$O(n^{2}m)$$ O ( n 2 m ) and $$O(n^{1+\omega })$$ O ( n 1 + ω ) , respectively. Additionally, based on a simplified graph traversal we propose a fast algorithm that runs in time $$O(n^2 + m^2)$$ O ( n 2 + m 2 ) and matches the corresponding running time of listing all simplicial vertices on sparse graphs with $$m=O(n)$$ m = O ( n ) . Moreover, we show that our algorithms cannot be improved significantly, as we prove that under plausible complexity assumptions there is no truly subquadratic algorithm for recognizing an avoidable vertex. To complement our results, we consider their natural generalizations of avoidable edges and avoidable paths. We propose an O(nm)-time algorithm that recognizes whether a given induced path is avoidable.

We present an Õ(m3/2) time randomized (Monte Carlo) algorithm for computing the 3-edge-connected components of a digraph with m edges and n vertices. This constitutes the first improvement since the algorithm of Nagamochi & Watanabe from 1993, which runs in O(m · n) time. Thus, our algorithm is the first that overcomes the run-time of O(n) computations of 3-bounded max-flows (that is, computations of the value min{Flow(s,t), 3} for O(n) pairs s-t). Our algorithm involves a combination of known and new techniques together with new structural insights on the interactions between directed min-cuts. One novel aspect that we introduce is an efficient graph operation G for replacing a set of vertices S that is disconnected from V\S by an edge-cut of size 2 (2-out set), with a gadget of small size that preserves the pairwise connectivity among the vertices of V\S. Another main ingredient of our approach is an extension of the framework for computing the vertex-connectivity (or edge-connectivity) in a digraph [Nanongkai et al., STOC'19]. This extension allows us to efficiently identify either all small 2-out sets of vertices, or identify enough 2-out sets whose total internal volume is a constant fraction of the edges of the graph. Repeatedly replacing each identified 2-out set S with a small gadget (using the G and G operations) either shrinks the size of the graph by a constant fraction, or concludes that no small 2-out set exists. We believe that our techniques may be of independent interest. Finally, we augment our algorithm with a data structure that can report in constant time the edges of some edge-cut of size at most 2 that disconnects any two query vertices u,v, or report in constant time that no such edge-cut exists.

Abstract The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $$2^{O(rw^3)}\cdot n^{4}$$ 2 O ( r w 3 ) · n 4 , $$2^{O(q^2\log (q))}\cdot n^{4}$$ 2 O ( q 2 log ( q ) ) · n 4 , and $$n^{O(k^2)}$$ n O ( k 2 ) where rw is the rank-width, q the $${\mathbb {Q}}$$ Q -rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on ( $$P_5$$ -free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of a polynomial-time algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of $$P_5$$ -free chordal graphs. Although the later result arises from the already-known reduction for $$P_5$$ -free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split graphs.

Motivated by the role of triadic closures in social networks, and the importance of finding a maximum subgraph avoiding a fixed pattern, we introduce and initiate the parameterized study of the StrongF-closure problem, where F is a fixed graph. This is a generalization of Strong Triadic Closure, whereas it is a relaxation of F-free Edge Deletion. In StrongF-closure, we want to select a maximum number of edges of the input graph G, and mark them as strong edges, in the following way: whenever a subset of the strong edges forms a subgraph isomorphic to F, then the corresponding induced subgraph of G is not isomorphic to F. Hence, the subgraph of G defined by the strong edges is not necessarily F-free, but whenever it contains a copy of F, there are additional edges in G to forbid that strong copy of F in G. We study StrongF-closure from a parameterized perspective with various natural parameterizations. Our main focus is on the number k of strong edges as the parameter. We show that the problem is FPT with this parameterization for every fixed graph F, whereas it does not admit a polynomial kernel even when $$F =P_3$$ F=P3. In fact, this latter case is equivalent to the Strong Triadic Closure problem, which motivates us to study this problem on input graphs belonging to well known graph classes. We show that Strong Triadic Closure does not admit a polynomial kernel even when the input graph is a split graph, whereas it admits a polynomial kernel when the input graph is planar, and even d-degenerate. Furthermore, on graphs of maximum degree at most 4, we show that Strong Triadic Closure is FPT with the above guarantee parameterization $$k - \mu (G)$$ k-μ(G), where $$\mu (G)$$ μ(G) is the maximum matching size of G. We conclude with some results on the parameterization of StrongF-closure by the number of edges of G that are not selected as strong.

In social networks the Strong Triadic Closure is an assignment of the edges with strong or weak labels such that any two vertices that have a common neighbor with a strong edge are adjacent. The problem of maximizing the number of strong edges that satisfy the strong triadic closure was recently shown to be NP-complete for general graphs. Here we initiate the study of graph classes for which the problem is solvable. We show that the problem admits a polynomial-time algorithm for two incomparable classes of graphs: proper interval graphs and trivially-perfect graphs. To complement our result, we show that the problem remains NP-complete on split graphs, and consequently also on chordal graphs. Thus, we contribute to define the first border between graph classes on which the problem is polynomially solvable and on which it remains NP-complete.

The (Weighted) Subset Feedback Vertex Set problem is a generalization of the classical Feedback Vertex Set problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although Subset Feedback Vertex Set and Feedback Vertex Set exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that Weighted Feedback Vertex Set can be solved in polynomial time. We provide a dichotomy result with respect to the size α of a maximum independent set. In particular we show that Weighted Subset Feedback Vertex Set can be solved in polynomial time for graphs with α≤3, whereas we prove that the problem remains NP-hard for graphs with α≥4. Moreover, we show that the (unweighted) Subset Feedback Vertex Set problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time nO(α). To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size. Node Multiway Cut is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for Subset Feedback Vertex Set, we settle the complexity of Node Multiway Cut as well as its variants where nodes are weighted and/or the terminals are deletable, for every value of the given independent set number.

Motivated by the emergence of large-scale networks in today's applications, we show how to compute efficiently smaller subgraphs that maintain some properties of an input graph. In particular, let G be a strongly connected directed graph. We consider the problem of computing the smallest strongly connected spanning subgraph of G that maintains certain connectivity relations of G. Specifically, for 2-edge-connectivity, we consider how to maintain the maximal 2-edge-connected subgraphs (2ECS) or the 2-edge-connected components (2ECC) of G, or both the maximal 2-edge-connected subgraphs and the 2-edge-connected components (2EC). Similarly, for 2-vertex-connectivity, we consider how to maintain the maximal 2-vertex-connected subgraphs (2VCS) or the 2-vertex-connected components (2VCC) of G, or both the maximal 2-vertex-connected subgraphs and the 2-vertex-connected components (2VC). All those problems are NP-hard, and thus we are interested in approximation algorithms. Additionally, we aim at designing algorithms with a good practical performance, so that they are able to scale effectively to very large graphs. While for 2ECS and 2VCS one can obtain an approximation ratio smaller than 2 by combining previously known results, providing good approximations for the 2-edge and the 2-vertex-components case seems more challenging. Here, we present linear-time approximation algorithms that achieve the following approximation guarantees: 4-approximation for 2ECC and 2EC, and 6-approximation for 2VCC and 2VC.

We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Threshold graphs are a well-studied subclass of interval graphs and of split graphs, both of which are unrelated subclasses of chordal graphs. To complement our result, we show that cutwidth is NP-complete on split graphs, and consequently also on chordal graphs. The cutwidth of interval graphs is still open, and only very few graph classes are known so far on which polynomial-time cutwidth algorithms exist. Thus we contribute to define the border between graph classes on which cutwidth is polynomially solvable and on which it remains NP-complete.