We prove that for any circle graph \(H\) with at least one edge and for any positive integer \(k\), there exists an integer \(t = t(k,H)\) so that every graph \(G\) either has a vertex-minor isomorphic to the disjoint union of \(k\) copies of \(H\), or has a \(t\)-perturbation with no vertex-minor isomorphic to \(H\). Using the same techniques, we also prove that for any planar multigraph \(H\), every binary matroid either has a minor isomorphic to the cycle matroid of \(kH\), or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of \(H\).

In this paper, we consider the class 𝒞^d of sphere intersection graphs in R^d for d ≥ 2. We show that for each integer t, the class of all graphs in 𝒞^d that exclude K_{t,t} as a subgraph has strongly sublinear separators. We also prove that 𝒞^d has asymptotic dimension at most 2d+2.

Abstract. We prove that if a graph contains the complete bipartite graph [Formula: see text] as an induced minor, then it contains a cycle of length at most 12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a graph contains [Formula: see text] as an induced minor, then it contains a triangle or a theta as an induced subgraph. Here, a theta is a graph made of three internally vertex-disjoint chordless paths [Formula: see text], [Formula: see text], [Formula: see text], each of length at least two, such that no edges exist between the paths except the three edges incident to [Formula: see text] and the three edges incident to [Formula: see text]. A consequence is that excluding a grid and a complete bipartite graph as induced minors is not enough to guarantee a bounded tree-independence number or even that the treewidth is bounded by a function of the size of the maximum clique, because the existence of graphs with large treewidth that contain no triangles or thetas as induced subgraphs is already known (the so-called layered wheels).

In 2015, Kawarabayashi and Kreutzer proved the directed grid theorem - the generalisation of the well-known excluded grid theorem to directed graphs - confirming a conjecture by Reed, Johnson, Robertson, Seymour, and Thomas from the mid-nineties. The theorem states the existence of a function$f$such that every digraph of directed tree-width$f(k)$contains a cylindrical grid of order$k$as a butterfly minor, but the given function grows non-elementarily with the size of the grid minor. More precisely, it contains a tower whose height depends on the size of the grid. In this paper, we present an alternative proof of the directed grid theorem which is conceptually much simpler, more modular in its composition and also improves the upper bound for the function$f$to a power tower of height 22. Our proof is inspired by the breakthrough result of Chekuri and Chuzhoy, who proved a polynomial bound for the excluded grid theorem for undirected graphs. We translate a key concept of their proof to directed graphs by introducing cycles of well-linked sets (CWS), and show that any digraph of high directed tree-width contains a large CWS, which in turn contains a large cylindrical grid, improving the result due to Kawarabayashi and Kreutzer from a non-elementary to an elementary function. An immediate application of our result is that we can improve the bound for Younger's conjecture-the directed Erdős-Pósa property-proved by Reed, Robertson, Seymour and Thomas [2] from a non-elementary to an elementary function. The same improvement applies to other types of Erdős-Pósa style problems on directed graphs. To the best of our knowledge, this is the first significant improvement on the bound for Younger's conjecture since it was proved in 1996. Since its publication in STOC 2015, the Directed Grid Theorem has found numerous applications (see for example [3]–[7]), all of which directly benefit from our main result. Finally, we believe that the theoretical tools developed in this work may find applications beyond the directed grid theorem, in a similar way as the path-of-sets-system framework due to Chekuri and Chuzhoy [8] did for undirected graphs (see for example [9]–[11]).

We show fixed-parameter tractability of the DIRECTED MULTICUT problem with three terminal pairs (with a randomized algorithm). In this problem we are given a directed graph G, three pairs of vertices (called terminals) (s1, t1), (s2, t2), (s3, t3), and an integer k and we want to find a set of at most k non-terminal vertices in G that intersect all s1t1-paths, all s2t2-paths, and all s3t3-paths. The parameterized complexity of this problem has been open since Chitnis, Hajiaghayi, and Marx proved fixed-parameter tractability of the two-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the four-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the “middle” box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced. * The full version of the paper can be accessed at https://arxiv.org/abs/2207.07425. The research leading to the results presented in this paper was partially carried out during the Parameterized Algorithms Retreat of the University of Warsaw, PARUW 2022, held in Bedlewo in April 2022. This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme Grant Agreement 714704 (TM, MP) and 648527 (MH), from the Alexander von Humboldt Foundation (MS), from the Research Council of Norway (LJ), and by the Federal Ministry of Education and Research (BMBF) and by a fellowship within the IFI programme of the German Academic Exchange Service (DAAD). (MH).

Moplexes are natural graph structures that arise when lifting Dirac’s classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes such as interval graphs, claw-free, and diamond-free graphs. However, while every non-complete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, among others, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are interval. In this work we initiate an investigation of k-moplex graphs, which are defined as graphs containing at most k moplexes. Of particular interest is the smallest nontrivial case, k = 2, which forms a counterpart to the class of interval graphs. As our main structural result, we show that the class of connected 2-moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely Graph Isomorphism and Max-Cut. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets. We also discuss sufficient conditions for the existence of Hamiltonian paths in 2-moplex graphs as well as connections with avoidable vertices.

The grid theorem, originally proved by Robertson and Seymour in 1986 [RS10, Graph Minors V], is one of the most central results in the study of graph minors and has found many algorithmic applications, especially in the analysis of routing problems. The relation between treewidth and grid minors is particularly tight for planar graphs, as every planar graph of treewidth at least 6k contains a grid of order k as a minor [RST94]. This polynomial, in fact linear, bound on the size of grid minors has enabled many important consequences, such as sublinear separators and subexponential algorithms for many NP-hard problems on planar graphs. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas proposed a notion of directed treewidth and conjectured an excluded grid theorem for directed graphs. This theorem was proved in 2015 [KK15] by the latter two authors but the function relating directed treewidth and grid minors is very big, even in the planar case. Directed grids have found several algorithmic applications such as low-congestion routing. See e.g. [CE15, CEP16, KKK14, EMW16, AKKW16]. However, in the undirected case the polynomial, in fact linear, bound on the size of grid minors in planar graphs have made this tool so extremely successful. Consequently, the lack of polynomial bounds for directed grid minors in planar digraphs has so far prevented further applications of this technique in the directed setting. The main result of this paper is to close this gap and to establish a polynomial bound for the directed grid theorem on planar digraphs. We are optimistic that this will enable further applications of directed treewidth and its dual notion of directed grids in the context of planar digraphs. Towards the end, we also give “treewidth sparsifier” for directed graphs, which has been already considered in undirected graphs. This result allows us to obtain an Eulerian subgraph of bounded degree in D that still has high directed treewidth. We believe this result is of independent interest for structural graph theory.