In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound.
This result is then used to prove the two following generalizations:
- every graph of treewidth less than $k$ that has a path of order $n$ contains an induced path of order at least $\frac{1}{4} (\log n)^{1/k}$;
- for every non-trivial graph class that is closed under topological minors there is a constant $d \in (0,1)$ such that every graph from this class that has a path of order $n$ contains an induced path of order at least $(\log n)^d$.
We also describe consequences of these results beyond graph classes that are closed under topological minors.
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.
Adjacency labelling for proper minor-closed graph classes
University of Ottawa（渥太华大学）；LaBRI, University of Bordeaux（波尔多大学 LaBRI）；Université libre de Bruxelles（布鲁塞尔自由大学）；Jagiellonian University（雅盖隆大学）；Carleton University（卡尔顿大学）；Monash University（蒙纳士大学）
For $p\in \mathbb{N}$, a coloring $λ$ of the vertices of a graph $G$ is {\em{$p$-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 $H$. In this paper, we prove that every $K_t$-minor-free graph admits a $p$-centered coloring with $\mathcal{O}(p^{g(t)})$ 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 $\mathcal{O}(p^{19})$ colors, with the degree of the polynomial independent of the genu…
Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k (\mathcal{G})$ the set of all graphs that are $k$-apices of $\mathcal{G}.$ We prove that every graph in the obstruction set of $\mathcal{A}_k (\mathcal{G}),$ i.e., the minor-minimal set of graphs not belonging to $\mathcal{A}_k (\mathcal{G}),$ has size at most $2^{2^{2^{2^{\mathsf{poly}(k)}}}},$ where $\mathsf{poly}$ is a polynomial function whose degree depends on…
Dujmović et al. [\emph{J.~ACM}~'20] recently proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, $p$-centered colouring, and adjacency labelling. This paper proves analogous product structure theorems for various non-minor-closed classes. One noteable example is $k$-planar graphs (those with a drawing in the plane in which each edge …
Obstructions for Minor-Closed Classes of limiting Densities Below 3/2
National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；Inter-Institutional Graduate Program “Algorithms, Logic, and Discrete Mathematics” (ALMA)（机构间研究生项目“算法、逻辑与离散数学”（ALMA））；NYUAD, CS-program（纽约大学阿布扎比分校计算机科学项目）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学国家科学研究中心计算机科学与微电子技术实验室）
The asymptotic dimension of metric spaces is an important notion in geometric group theory introduced by Gromov. The metric spaces considered in this paper are the ones whose underlying spaces are the vertex-sets of graphs and whose metrics are the distance functions in graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite graphs. In this paper we prove that the asymptotic dimension of any proper minor-closed family, any class of graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, …
Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $(tw,ω)$-bounded. Such graph classes are known to have useful algorithmic applications related to variants of the clique and $k$-coloring problems. We consider six well-known graph containment relations: the minor, topological minor, subgraph, induced minor, induced topological minor,…
Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph $G$ belongs to a given graph class~$\mathcal{G}$. Such certifications with labels of size $O(\log n)$ (where $n$ is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite se…
A graph is an apex graph if it contains a vertex whose deletion leaves a planar graph. The family of apex graphs is minor-closed and so it is characterized by a finite list of minor-minimal non-members. The long-standing problem of determining this finite list of apex obstructions remains open. This paper determines the 133 minor-minimal, non-apex graphs that have connectivity two.
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 $β$ with $0<β<1$, if $\mathcal G$ is a grap…
Nowhere dense graph classes and algorithmic applications. A tutorial at Highlights of Logic, Games and Automata 2019
无处稠密图类的概念是由Nešetzil和Ossona de Mendez提出的，它提供了一个鲁棒的图类一致稀疏的概念。无处密集类推广了许多熟悉的类的稀疏图，如类，排除一个固定的图作为一个小的或拓扑的小。他们承认几个看似无关的自然特征，导致强大的算法应用。特别是，一阶逻辑的模型检查问题是固定参数的这些类的易处理的。这些笔记是为2019年逻辑，游戏和自动机亮点的教程准备的，是对算法应用驱动的无处密集理论的简要介绍。
It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various non-minor-closed classes, and graph classes with polynomial growth. We then explore how graph product structure might be applicable to more broadly defined graph classes. In particular, we characterise when a graph class defined by a cartesian or strong product has bounded or polynomial expansion. We then explore graph product structure theorems for various geometr…
Domination Problems in Nowhere-Dense Classes of Graphs
我们调查的参数化复杂性的概括和变化的支配集问题的类图无处密集。特别是，我们证明了距离d的支配集问题，也被称为（k，d）-中心问题，是固定参数听话的任何类，无处密集和封闭的诱导子图。这推广了已知的结果支配集问题的H-未成年人自由类，类与局部排除未成年人和类的图的有界扩张。我们的证明的一个关键特征是，它是简单地基于这样一个事实，即这些图类是一致的准宽，并不依赖于结构分解。我们的结果还建立了距离d的支配集问题是FPT类的有界扩展，回答了一个问题Ne{ \v s}et{ziness}il和Ossona de Mendez。
A Logic for Minor-Free Graph Classes: Model Checking, Dependence, and Combinatorial Reconfiguration
University of Warsaw（华沙大学）；CAMS, EHESS/CNRS UMR 8557（EHESS/CNRS联合研究单位8557）；University of Bremen（不来梅大学）；Université Paris Cité, CNRS, IRIF（巴黎西岱大学，法国国家科学研究中心，信息学研究所）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学，法国国家科学研究中心，计算机科学与微电子研究所）；LIMOS, Univ Clermont Auvergne, Clermont Auvergne INP（克勒蒙奥弗涅大学，克勒蒙奥弗涅理工学院，计算与信息系统实验室）
