Minor-closed Class Research Articles

Minor-closed classes of binary functions

Binary functions are a generalisation of the cocircuit spaces of binary matroids to arbitrary functions. Every rank function is assigned a binary function, and the deletion and contraction operations of binary functions generalise matroid deletion and contraction. We give the excluded minor characterisations for the classes of binary functions with well defined minors, and those with an associated rank function. Within these classes, we also characterise the classes of binary functions corresponding to polymatroids, matroids and binary matroids by their excluded minors. This gives a new proof of Tutte's excluded minor characterisation of binary matroids in the more generalised space of binary functions.

Faster parameterized algorithms for modification problems to minor-closed classes

Let ${\cal G}$ be a minor-closed graph class and let $G$ be an $n$-vertex graph. We say that $G$ is a $k$-apex of ${\cal G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to ${\cal G}$. Our first result is an algorithm that decides whether $G$ is a $k$-apex of ${\cal G}$ in time $2^{{\sf poly}(k)}\cdot n^2$, where ${\sf poly}$ is a polynomial function depending on ${\cal G}$. This algorithm improves the previous one, given by Sau, Stamoulis, and Thilikos [ICALP 2020], whose running time was $2^{{\sf poly}(k)}\cdot n^3$. The elimination distance of $G$ to ${\cal G}$, denoted by ${\sf ed}_{\cal G}(G)$, is the minimum number of rounds required to reduce each connected component of $G$ to a graph in ${\cal G}$ by removing one vertex from each connected component in each round. Bulian and Dawar [Algorithmica 2017] provided an FPT-algorithm, with parameter $k$, to decide whether ${\sf ed}_{\cal G}(G)\leq k$. However, its dependence on $k$ is not explicit. We extend the techniques used in the first algorithm to decide whether ${\sf ed}_{\cal G}(G)\leq k$ in time $2^{2^{2^{{\sf poly}(k)}}}\cdot n^2$. This is the first algorithm for this problem with an explicit parametric dependence in $k$. In the special case where ${\cal G}$ excludes some apex-graph as a minor, we give two alternative algorithms, running in time $2^{2^{{\cal O}(k^2\log k)}}\cdot n^2$ and $2^{{\sf poly}(k)}\cdot n^3$ respectively, where $c$ and ${\sf poly}$ depend on ${\cal G}$. As a stepping stone for these algorithms, we provide an algorithm that decides whether ${\sf ed}_{\cal G}(G)\leq k$ in time $2^{{\cal O}({\sf tw}\cdot k+{\sf tw}\log{\sf tw})}\cdot n$, where ${\sf tw}$ is the treewidth of $G$. Finally, we provide explicit upper bounds on the size of the graphs in the minor-obstruction set of the class of graphs ${\cal E}_k({\cal G})=\{G\mid{\sf ed}_{\cal G}(G)\leq k\}$.

Killing a Vortex

Killing a Vortex

On the parameterized complexity of the acyclic matching problem

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to the parameters treewidth, modular-width and (k,c4), where c4 is the number of cycles with length 4 of the input graph. We also prove that the problem is fixed parameter tractable with respect to the parameter k for the line graphs and every proper minor-closed class of graphs (including planar graphs).

Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k

Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field F, the list needs to contain only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J.F. Geelen, A.M.H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F-representable excluded minors in general.We consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2|F|O(k2) elements. We can therefore compute, for any integer k and a fixed finite field F, the set of F-representable excluded minors for the class of matroids of path-width k, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F-represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 22O(k2) vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.

Clustered colouring of graph classes with bounded treedepth or pathwidth

AbstractThe 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 determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.

The Excluded Minors for Lattice Path Polymatroids

We find the excluded minors for the minor-closed class of lattice path polymatroids as a subclass of the minor-closed class of Boolean polymatroids. Like lattice path matroids and Boolean polymatroids, there are infinitely many excluded minors, but they fall into a small number of easily-described types.

Separating layered treewidth and row treewidth

Layered treewidth and row treewidth are recently introduced graph parameters that have been key ingredients in the solution of several well-known open problems. It follows from the definitions that the layered treewidth of a graph is at most its row treewidth plus 1. Moreover, a minor-closed class has bounded layered treewidth if and only if it has bounded row treewidth. However, it has been open whether row treewidth is bounded by a function of layered treewidth. This paper answers this question in the negative. In particular, for every integer $k$ we describe a graph with layered treewidth 1 and row treewidth $k$. We also prove an analogous result for layered pathwidth and row pathwidth.

Subgraph densities in a surface

AbstractGiven a fixed graphHthat embeds in a surface$\Sigma$, what is the maximum number of copies ofHin ann-vertex graphGthat embeds in$\Sigma$? We show that the answer is$\Theta(n^{f(H)})$, wheref(H) is a graph invariant called the ‘flap-number’ ofH, which is independent of$\Sigma$. This simultaneously answers two open problems posed by Eppstein ((1993)J. Graph Theory17(3) 409–416.). The same proof also answers the question for minor-closed classes. That is, ifHis a$K_{3,t}$minor-free graph, then the maximum number of copies ofHin ann-vertex$K_{3,t}$minor-free graph G is$\Theta(n^{f'(H)})$, wheref′(H) is a graph invariant closely related to the flap-number ofH. Finally, whenHis a complete graph we give more precise answers.

On the Parameterized Complexity Of the Acyclic Matching Problem

A matching is a set of edges in a graph with no common endpoint. A matching $M$ is called acyclic if the induced subgraph on the endpoints of the edges in $M$ is acyclic. Given a graph $G$ and an integer $k$, Acyclic Matching Problem seeks for an acyclic matching of size $k$ in $G$. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter $k$. On the other hand, the problem is fixed parameter tractable with respect to $k$, for line graphs, $C_4$-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).

Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

Twin-width I: Tractable FO Model Checking

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

Influence of Auditory-Verbal Therapy in Speech and Language Development in Cuban Children With Pre-Linguistic Sensorineural Hearing Loss With Cochlear Implants: A Longitudinal Study

Motivated by a conjecture of Gyárfás, recently Böttcher, Hladký, Piguet, and Taraz showed that every collection T1,…,Tt of trees on n vertices with ∑i=1te(Ti)⩽(n2) and with bounded maximum degree can be packed into the complete graph on (1+o(1))n vertices. We generalise this result where we relax the restriction of packing families of trees to families of graphs of any given non-trivial minor-closed class of graphs.

Polynomial bounds for centered colorings on proper minor-closed graph classes

For p∈N, a coloring λ of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. Centered colorings play an important role in the theory of sparse graph classes introduced by Nešetřil and Ossona de Mendez [31,32], as they structurally characterize classes of bounded expansion — one of the key sparsity notions in this theory. More precisely, a class of graphs C has bounded expansion if and only if there is a function f:N→N such that every graph G∈C for every p∈N admits a p-centered coloring with at most f(p) colors. Unfortunately, known proofs for the existence of such colorings yield large upper bounds on the function f governing the number of colors needed, even for as simple classes as planar graphs. In this paper, we prove that every Kt-minor-free graph admits a p-centered coloring with O(pg(t)) colors for some function g. In the special case that the graph is embeddable in a fixed surface Σ we show that it admits a p-centered coloring with O(p19) colors, with the degree of the polynomial independent of the genus of Σ. This provides the first polynomial upper bounds on the number of colors needed in p-centered colorings of graphs drawn from proper minor-closed classes, which answers an open problem posed by Dvořák [1]. As an algorithmic application, we use our main result to prove that if C is a fixed proper minor-closed class of graphs, then given graphs H and G, on p and n vertices, respectively, where G∈C, it can be decided whether H is a subgraph of G in time 2O(plog⁡p)⋅nO(1) and space nO(1).

Unavoidable Minors for Graphs with Large $$\ell _p$$-Dimension

A metric graph is a pair (G, d), where G is a graph and \(d:E(G) \rightarrow \mathbb {R}_{\ge 0}\) is a distance function. Let \(p \in [1,\infty ]\) be fixed. An isometric embedding of the metric graph (G, d) in \(\ell _p^k = (\mathbb {R}^k, d_p)\) is a map \(\phi :V(G) \rightarrow \mathbb {R}^k\) such that \(d_p(\phi (v), \phi (w)) = d(vw)\) for all edges \(vw\in E(G)\). The \(\ell _p\)-dimension of G is the least integer k such that there exists an isometric embedding of (G, d) in \(\ell _p^k\) for all distance functions d such that (G, d) has an isometric embedding in \(\ell _p^K\) for some K. It is easy to show that \(\ell _p\)-dimension is a minor-monotone property. In this paper, we characterize the minor-closed graph classes \(\mathscr {C}\) with bounded \(\ell _p\)-dimension, for \(p \in \{2,\infty \}\). For \(p=2\), we give a simple proof that \(\mathscr {C}\) has bounded \(\ell _2\)-dimension if and only if \(\mathscr {C}\) has bounded treewidth. In this sense, the \(\ell _2\)-dimension of a graph is ‘tied’ to its treewidth. For \(p=\infty \), the situation is completely different. Our main result states that a minor-closed class \(\mathscr {C}\) has bounded \(\ell _\infty \)-dimension if and only if \(\mathscr {C}\) excludes a graph obtained by joining copies of \(K_4\) using the 2-sum operation, or excludes a Möbius ladder with one ‘horizontal edge’ removed.

Additive non-approximability of chromatic number in proper minor-closed classes

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally, for every k≥1 and 1≤β≤4/3, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4−3β)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes.Furthermore, we give somewhat weaker non-approximability bound for K4k+1-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.

Planar Graphs Have Bounded Queue-Number

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] from 1992. The key to the proof is a new structural tool called layered partitions , and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. [31] that graphs in a proper minor-closed class have low treewidth colourings.

Decomposable polymatroids and connections with graph coloring

We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of rank functions of matroids, and for which the minimum number of matroids required in such sums is the chromatic number of the line graph of the hypergraph. This result motivates introducing chromatic numbers and chromatic polynomials for polymatroids. We show that the chromatic polynomial of any 2-polymatroid is a rational multiple of the chromatic polynomial of some graph. We also find the excluded minors for the minor-closed class of polymatroids that can be written as sums of rank functions of matroids that form a chain of quotients.

Fractal classes of matroids

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k≥5) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.

The Matroid Secretary Problem for Minor-Closed Classes and Random Matroids

We prove that for every proper minor-closed class $\mathcal{M}$ of $\mathbb{F}_p$-representable matroids, there exists an $O(1)$-competitive algorithm for the matroid secretary problem on $\mathcal...