Minor-closed Class Research Articles

Minor-Closed Graph Classes with Bounded Layered Pathwidth

We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class.

The Weisfeiler--Leman Dimension of Planar Graphs Is at Most 3

We prove that the Weisfeiler--Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best-known upper bounds for the dimension and number of variables were 14 and 15, respectively. First, we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of every arc-colored 3-connected graph belonging to this class. Then, we prove that, apart from several exceptional graphs (which have WL-dimension at most 2), the individualization of two appropriately chosen vertices of a colored 3-connected planar graph followed by the one-dimensional WL-algorithm produces the discrete vertex partition. This implies that the three-dimensional WL-algorithm determines the orbits of arc-colored 3-connected planar graphs. As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3.

Clustered Colouring in Minor-Closed Classes

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We prove that for every graph $H$, the clustered chromatic number of the class of $H$-minor-free graphs is tied to the tree-depth of $H$. In particular, if $H$ is connected with tree-depth $t$ then every $H$-minor-free graph is $(2^{t+1}-4)$-colourable with monochromatic components of size at most $c(H)$. This provides the first evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about defective colouring of $H$-minor-free graphs. If $t=3$ then we prove that 4 colours suffice, which is best possible. We also determine those minor-closed graph classes with clustered chromatic number 2. Finally, we develop a conjecture for the clustered chromatic number of an arbitrary minor-closed class.

The Möbius matroids

In three influential papers in the 1980s and early 1990s, Joe Kung laid the foundations for extremal matroid theory which he envisaged as finding the growth rate of certain classes of matroids along with a characterisation of the extremal matroids in each such class. At the time, he was particularly interested in the minor-closed classes of binary matroids obtained by excluding the cycle matroids of the Kuratowski graphs K3,3 and/or K5. While he obtained strong bounds on the growth rate of these classes, it seems difficult to give the exact growth rate without a complete characterisation of the matroids in each class, which at the time seemed hopelessly complicated. Many years later, Mayhew, Royle and Whittle gave a characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor, from which the answers to Kung's questions follow immediately. In this characterisation, two thin families of binary matroids play an unexpectedly important role as the only non-cographic infinite families of internally 4-connected binary matroids with no M(K3,3)-minor. As the matroids are closely related to the cubic and quartic Möbius ladders, they were called the triangular Möbius matroids and the triadic Möbius matroids. Preliminary investigations of the class of binary matroids with no M(K5)-minor suggest that, once again, the triangular Möbius matroids will be the extremal internally 4-connected matroids in this class. Here we undertake a systematic study of these two families of binary matroids collecting in one place fundamental information about them, including their representations, connectivity properties, minor structure, automorphism groups and their chromatic polynomials. Along the way, we highlight the different ways in which these matroids have arisen naturally in a number of results and problems (both open and settled) in structural and extremal matroid theory.

Delta-matroids as subsystems of sequences of Higgs lifts

In [30], Tardos studied special delta-matroids obtained from sequences of Higgs lifts; these are the full Higgs lift delta-matroids that we treat and around which all of our results revolve. We give an excluded-minor characterization of the class of full Higgs lift delta-matroids within the class of all delta-matroids, and we give similar characterizations of two other minor-closed classes of delta-matroids that we define using Higgs lifts. We introduce a minor-closed, dual-closed class of Higgs lift delta-matroids that arise from lattice paths. It follows from results of Bouchet that all delta-matroids can be obtained from full Higgs lift delta-matroids by removing certain feasible sets; to address which feasible sets can be removed, we give an excluded-minor characterization of delta-matroids within the more general structure of set systems. Many of these excluded minors occur again when we characterize the delta-matroids in which the collection of feasible sets is the union of the collections of bases of matroids of different ranks, and yet again when we require those matroids to have special properties, such as being paving.

Extension complexity of the correlation polytope

We prove that for every n-vertex graph G, the extension complexity of the correlation polytope of G is 2O(tw(G)+logn), where tw(G) is the treewidth of G. Our main result is that this bound is tight for graphs contained in minor-closed classes.

On the critical densities of minor-closed classes

Given a minor-closed class A of graphs, let βA denote the supremum over all graphs in A of the ratio of edges to vertices. We investigate the set B of all such values βA, taking further the project begun by Eppstein. Amongst other results, we determine the small values in B (those up to 2); we show that B is ‘asymptotically dense’; and we answer some questions posed by Eppstein.

On the purity of minor-closed classes of graphs

Given a graph H with at least one edge, let gapH(n) denote the maximum difference between the numbers of edges in two n-vertex edge-maximal graphs with no minor H. We show that for exactly four connected graphs H (with at least two vertices), the class of graphs with no minor H is pure, that is, gapH(n)=0 for all n⩾1; and for each connected graph H (with at least two vertices) we have the dichotomy that either gapH(n)=O(1) or gapH(n)=Θ(n). Further, if H is 2-connected and does not yield a pure class, then there is a constant c>0 such that gapH(n)∼cn. We also give some partial results when H is not connected or when there are two or more excluded minors.

Reflections on and of minor-closed classes of multisorted operations

The minor relation of functions is generalized to multisorted functions. Pippenger’s Galois theory for minor-closed sets of functions is extended to multisorted functions and multisorted relation pairs. Reflections of minor-closed sets are again minor-closed, and the effect of reflections on the invariant relation pairs of minor-closed sets of multisorted functions is described.

On Perturbations of Highly Connected Dyadic Matroids

Geelen, Gerards, and Whittle [3] announced the following result: let $q = p^k$ be a prime power, and let $\mathcal{M}$ be a proper minor-closed class of $\mathrm{GF}(q)$-representable matroids, which does not contain $\mathrm{PG}(r-1,p)$ for sufficiently high $r$. There exist integers $k, t$ such that every vertically $k$-connected matroid in $\mathcal{M}$ is a rank-$(\leq t)$ perturbation of a frame matroid or the dual of a frame matroid over $\mathrm{GF}(q)$. They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates. We show a family of dyadic matroids that form a counterexample to this result. We offer several weaker conjectures to replace the ones in [3], discuss consequences for some published papers, and discuss the impact of these new conjectures on the structure of frame templates.

Logical limit laws for minor-closed classes of graphs

Let G be an addable, minor-closed class of graphs. We prove that the zero-one law holds in monadic second-order logic (MSO) for the random graph drawn uniformly at random from all connected graphs in G on n vertices, and the convergence law in MSO holds if we draw uniformly at random from all graphs in G on n vertices. We also prove analogues of these results for the class of graphs embeddable on a fixed surface, provided we restrict attention to first order logic (FO). Moreover, the limiting probability that a given FO sentence is satisfied is independent of the surface S. We also prove that the closure of the set of limiting probabilities is always the finite union of at least two disjoint intervals, and that it is the same for FO and MSO. For the classes of forests and planar graphs we are able to determine the closure of the set of limiting probabilities precisely. For planar graphs it consists of exactly 108 intervals, each of the same length ≈5.39⋅10−6. Finally, we analyse examples of non-addable classes where the behaviour is quite different. For instance, the zero-one law does not hold for the random caterpillar on n vertices, even in FO.

On excluded minors for classes of graphical matroids

Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let M be one of these three classes, and let r be a positive integer. We show that M has only a finite number of excluded minors of rank r.

Stack and Queue Layouts via Layered Separators

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case of $1$-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that $n$-vertex graphs that admit constant layered separators have $\mathcal{O}(\log n)$ stack-number.

Orthogonal Tree Decompositions of Graphs

This paper studies graphs that have two tree decompositions with the property that every bag from the first decomposition has a bounded-size intersection with every bag from the second decomposition. We show that every graph in each of the following classes has a tree decomposition and a linear-sized path decomposition with bounded intersections: (1) every proper minor-closed class, (2) string graphs with a linear number of crossings in a fixed surface, (3) graphs with linear crossing number in a fixed surface. Here `linear size' means that the total size of the bags in the path decomposition is $O(n)$ for $n$-vertex graphs. We then show that every $n$-vertex graph that has a tree decomposition and a linear-sized path decomposition with bounded intersections has $O(\sqrt{n})$ treewidth. As a corollary, we conclude a new lower bound on the crossing number of a graph in terms of its treewidth. Finally, we consider graph classes that have two path decompositions with bounded intersections. Trees and outerplanar graphs have this property. But for the next most simple class, series parallel graphs, we show that no such result holds.

Layered separators in minor-closed graph classes with applications

Graph separators are a ubiquitous tool in graph theory and computer science. However, in some applications, their usefulness is limited by the fact that the separator can be as large as Ω(n) in graphs with n vertices. This is the case for planar graphs, and more generally, for proper minor-closed classes. We study a special type of graph separator, called a layered separator, which may have linear size in n, but has bounded size with respect to a different measure, called the width. We prove, for example, that planar graphs and graphs of bounded Euler genus admit layered separators of bounded width. More generally, we characterise the minor-closed classes that admit layered separators of bounded width as those that exclude a fixed apex graph as a minor.We use layered separators to prove O(log⁡n) bounds for a number of problems where O(n) was a long-standing previous best bound. This includes the nonrepetitive chromatic number and queue-number of graphs with bounded Euler genus. We extend these results with a O(log⁡n) bound on the nonrepetitive chromatic number of graphs excluding a fixed topological minor, and a logO(1)⁡n bound on the queue-number of graphs excluding a fixed minor. Only for planar graphs were logO(1)⁡n bounds previously known. Our results imply that every n-vertex graph excluding a fixed minor has a 3-dimensional grid drawing with nlogO(1)⁡n volume, whereas the previous best bound was O(n3/2).

The densest matroids in minor-closed classes with exponential growth rate

The growth rate function for a nonempty minor-closed class of matroids M \mathcal {M} is the function h M ( n ) h_{\mathcal {M}}(n) whose value at an integer n ≥ 0 n \ge 0 is defined to be the maximum number of elements in a simple matroid in M \mathcal {M} of rank at most n n . Geelen, Kabell, Kung and Whittle showed that, whenever h M ( 2 ) h_{\mathcal {M}}(2) is finite, the function h M h_{\mathcal {M}} grows linearly, quadratically or exponentially in n n (with base equal to a prime power q q ), up to a constant factor. We prove that in the exponential case, there are nonnegative integers k k and d ≤ q 2 k − 1 q − 1 d \le \tfrac {q^{2k}-1} {q-1} such that h M ( n ) = q n + k − 1 q − 1 − q d h_{\mathcal {M}}(n) = \frac {q^{n+k}-1}{q-1} - qd for all sufficiently large n n , and we characterise which matroids attain the growth rate function for large n n . We also show that if M \mathcal {M} is specified in a certain ‘natural’ way (by intersections of classes of matroids representable over different finite fields and/or by excluding a finite set of minors), then the constants k k and d d , as well as the point that ‘sufficiently large’ begins to apply to n n , can be determined by a finite computation.

Infinitely many minimal classes of graphs of unbounded clique-width

The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width.

Inclusion of forbidden minors in random representable matroids

In 1984, Kelly and Oxley introduced the model of a random representable matroid M[An] corresponding to a random matrix An∈Fqm(n)×n, whose entries are drawn independently and uniformly from Fq. Whereas properties such as rank, connectivity, and circuit size have been well-studied, forbidden minors have not yet been analyzed. Here, we investigate the asymptotic probability as n→∞ that a fixed Fq-representable matroid M is a minor of M[An]. (We always assume m(n)≥rank(M) for all sufficiently large n, otherwise M can never be a minor of the corresponding M[An].) When M is free, we show that M is asymptotically almost surely (a.a.s.) a minor of M[An]. When M is not free, we show a phase transition: M is a.a.s. a minor if n−m(n)→∞, but is a.a.s. not if m(n)−n→∞. In the more general settings of m≤n and m>n, we give lower and upper bounds, respectively, on both the asymptotic and non-asymptotic probabilities that M is a minor of M[An]. The tools we develop to analyze matroid operations and minors of random matroids may be of independent interest.Our results directly imply that M[An] is a.a.s. not contained in any proper, minor-closed class M of Fq-representable matroids, provided: (i) n−m(n)→∞, and (ii) m(n) is at least the minimum rank of any Fq-representable forbidden minor of M, for all sufficiently large n. As an application, this shows that graphic matroids are a vanishing subset of linear matroids, in a sense made precise in the paper. Our results provide an approach for applying the rich theory around matroid minors to the less-studied field of random matroids.

The Extremal Functions of Classes of Matroids of Bounded Branch-Width

For a set of matroids M, let ex M (n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field $$\mathbb{F}$$ , if M is a minor-closed class of $$\mathbb{F}$$ -representable matroids of bounded branch-width, then limn→ ∞ex M (n)/n exists and is a rational number, ∆. We also show that ex M (n) - ∆n is periodic when n is sufficiently large and that exM is achieved by a subclass of M of bounded path-width.

Templates for Binary Matroids

A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a preorder on these templates and determine the nontrivial templates that are minimal with respect to this order. As an application of our main result, we determine the eventual growth rates of certain minor-closed classes of binary matroids, including the class of binary matroids with no minor isomorphic to PG(3,2). Our main result applies to all highly-connected matroids in a class, not just those of maximum size. As a second application, we characterize the highly-connected 1-flowing matroids.