Minor-free Graphs Research Articles

Approximation schemes for Min-Sum [formula omitted]-Clustering

We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G=(V,E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz et al. (2019) achieve an approximation ratio of O(log|V|) in polynomial time for general metrics and an approximation ratio 2+ϵ in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics.

Reducing linear Hadwiger’s conjecture to coloring small graphs

In 1943, Hadwiger conjectured that every graph with no K t K_t minor is ( t − 1 ) (t-1) -colorable for every t ≥ 1 t\ge 1 . In the 1980s, Kostochka and Thomason independently proved that every graph with no K t K_t minor has average degree O ( t log ⁡ t ) O(t\sqrt {\log t}) and hence is O ( t log ⁡ t ) O(t\sqrt {\log t}) -colorable. Recently, Norin, Song and the second author showed that every graph with no K t K_t minor is O ( t ( log ⁡ t ) β ) O(t(\log t)^{\beta }) -colorable for every β > 1 / 4 \beta > 1/4 , making the first improvement on the order of magnitude of the O ( t log ⁡ t ) O(t\sqrt {\log t}) bound. The first main result of this paper is that every graph with no K t K_t minor is O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. This is actually a corollary of our main technical result that the chromatic number of a K t K_t -minor-free graph is bounded by O ( t ( 1 + f ( G , t ) ) ) O(t(1+f(G,t))) where f ( G , t ) f(G,t) is the maximum of χ ( H ) a \frac {\chi (H)}{a} over all a ≥ t log ⁡ t a\ge \frac {t}{\sqrt {\log t}} and K a K_a -minor-free subgraphs H H of G G that are small (i.e. O ( a log 4 ⁡ a ) O(a\log ^4 a) vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small K t K_t -minor-free graphs, we show that K t K_t -minor-free graphs are O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. Second, it shows that proving Linear Hadwiger’s Conjecture (that K t K_t -minor-free graphs are O ( t ) O(t) -colorable) reduces to proving it for small graphs. Third, we prove that K t K_t -minor-free graphs with clique number at most log ⁡ t / ( log ⁡ log ⁡ t ) 2 \sqrt {\log t}/ (\log \log t)^2 are O ( t ) O(t) -colorable. This implies our final corollary that Linear Hadwiger’s Conjecture holds for K r K_r -free graphs for every fixed r r ; more generally, we show there exists C ≥ 1 C\ge 1 such that for every r ≥ 1 r\ge 1 , there exists t r t_r such that for all t ≥ t r t\ge t_r , every K r K_r -free K t K_t -minor-free graph is C t Ct -colorable. One key to proving the main theorem is building the minor in two new ways according to whether the chromatic number ‘separates’ in the graph: sequentially if the graph is the chromatic-inseparable and recursively if the graph is chromatic-separable. The other key is a new standalone result that every K t K_t -minor-free graph of average degree d = Ω ( t ) d=\Omega (t) has a subgraph on O ( t log 3 ⁡ t ) O(t \log ^3 t) vertices with average degree Ω ( d ) \Omega (d) .

Good r-divisions Imply Optimal Amortized Decremental Biconnectivity

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(m+n)$$\\end{document} time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

Killing a Vortex

Killing a Vortex

Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs

AbstractGiven a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC $_{con}$ . In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$ -time FPT algorithm for MinCkSC $_{con}$ on a graph with treewidth $\omega$ , based on which we further design an $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC $_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.

Product structure of graph classes with bounded treewidth

AbstractWe show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$ , for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$ . We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$ ; the class of $K_{s,t}$ -minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$ ); and the class of $K_t$ -minor-free graphs has underlying treewidth $t-2$ . In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

The spectral radius of minor-free graphs

In this paper, we present a sharp upper bound for the spectral radius of an n-vertex F-minor-free graph for sufficiently large n, where F is obtained from the complete graph Kr by deleting disjoint paths. Furthermore, the graphs which achieve the sharp bound are characterized. This result may be regarded as a spectral extremal analogue of the number of edges in an n-vertex F-minor-free graph.

On the choosability of -minor-free graphs

AbstractGiven a graph $H$ , let us denote by $f_\chi (H)$ and $f_\ell (H)$ , respectively, the maximum chromatic number and the maximum list chromatic number of $H$ -minor-free graphs. Hadwiger’s famous colouring conjecture from 1943 states that $f_\chi (K_t)=t-1$ for every $t \ge 2$ . A closely related problem that has received significant attention in the past concerns $f_\ell (K_t)$ , for which it is known that $2t-o(t) \le f_\ell (K_t) \le O(t (\!\log \log t)^6)$ . Thus, $f_\ell (K_t)$ is bounded away from the conjectured value $t-1$ for $f_\chi (K_t)$ by at least a constant factor. The so-called $H$ -Hadwiger’s conjecture, proposed by Seymour, asks to prove that $f_\chi (H)={\textrm{v}}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger’s conjecture).In this paper, we prove several new lower bounds on $f_\ell (H)$ , thus exploring the limits of a list colouring extension of $H$ -Hadwiger’s conjecture. Our main results are:For every $\varepsilon \gt 0$ and all sufficiently large graphs $H$ we have $f_\ell (H)\ge (1-\varepsilon )({\textrm{v}}(H)+\kappa (H))$ , where $\kappa (H)$ denotes the vertex-connectivity of $H$ .For every $\varepsilon \gt 0$ there exists $C=C(\varepsilon )\gt 0$ such that asymptotically almost every $n$ -vertex graph $H$ with $\left \lceil C n\log n\right \rceil$ edges satisfies $f_\ell (H)\ge (2-\varepsilon )n$ .The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$ -minor-free graphs is separated from the desired value of $({\textrm{v}}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that for almost all graphs $H$ with superlogarithmic average degree $f_\ell (H)$ is separated from $({\textrm{v}}(H)-1)$ by a constant factor arbitrarily close to $2$ . Conceptually these results indicate that the graphs $H$ for which $f_\ell (H)$ is close to the conjectured value $({\textrm{v}}(H)-1)$ for $f_\chi (H)$ are typically rather sparse.

On the bond polytope

Abstract While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural closely related variant maximum bond. Here, given a graph G = (V, E), we ask for a maximum cut δ(S) ⊆ E with S ⊆ V under the restriction that both G[S] as well as G[V \ S] are connected. Observe that both the maximum cut and the maximum bond can be seen as inverse problems to the traditional minimum cut, as there, the connectivity arises naturally in optimal solutions. The bond polytope is the convex hull of all incidence vectors of bonds. Similar to the connection of the corresponding optimization problems, the bond polytope is closely related to the cut polytope. While the latter has been intensively studied, there are no results on bond polytopes. We start a structural study of the latter, which additionally allows us to deduce algorithmic consequences. We investigate the relation between cut- and bond polytopes and the additional intricacies that arise when requiring connectivity in the solutions. We study the effect of graph modifications on bond polytopes and their facets, akin to what has been spearheaded for cut polytopes by Barahona, Grötschel and Mahjoub [4; 3] and Deza and Laurant [17; 15; 16]. Moreover, we study facet-defining inequalities arising from edges and cycles for bond polytopes. In particular, these yield a complete linear description of bond polytopes of cycles and 3-connected planar (K 5 − e)-minor free graphs. Finally, we present a reduction of the maximum bond problem on arbitrary graphs to the maximum bond problem on 3-connected graphs. This yields a linear time algorithm for maximum bond on (K 5 − e)-minor free graphs.

Extended Double Covers and Homomorphism Bounds of Signed Graphs

A signed graph $(G, \sigma)$ is a graph $G$ together with an assignment $\sigma:E(G) \rightarrow \{+,-\}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between the theories of minors and colorings of graphs. Following this thread of thoughts, we investigate this connection through the notion of Extended Double Covers of signed graphs, which was recently introduced by Naserasr, Sopena and Zaslavsky. More precisely, we say that a signed graph $(B, \pi)$ is planar-complete if any signed planar graph $(G, \sigma)$ which verifies the conditions of a basic no-homomorphism lemma with respect to $(B,\pi)$ admits a homomorphism to $(B, \pi)$. Our conjecture then is that: if $(B, \pi)$ is a connected signed graph with no positive odd closed walk which is planar-complete, then its Extended Double Cover ${\rm EDC}(B,\pi)$ is also planar-complete. We observe that this conjecture largely extends the Four-Color Theorem and is strongly connected to a number of conjectures in extension of this famous theorem. A given (signed) graph $(B,\pi)$ bounds a class of (signed) graphs if every (signed) graph in the class admits a homomorphism to $(B,\pi)$.In this work, and in support of our conjecture, we prove it for the subclass of signed $K_4$-minor free graphs. Inspired by this development, we then investigate the problem of finding optimal homomorphism bounds for subclasses of signed $K_4$-minor-free graphs with restrictions on their girth and we present nearly optimal solutions. Our work furthermore leads to the development of weighted signed graphs.

The linear arboricity of K5-minor free graphs

The linear arboricity l a ( G ) of a graph G is the minimum number of linear forests which partition the edges of G . In 1980, Akiyama et al. conjectured that for any graph G , ⌈ Δ ( G ) 2 ⌉ ≤ l a ( G ) ≤ ⌈ Δ ( G ) + 1 2 ⌉ . In the paper, we establish two essential structural properties of K 5 -minor free graphs G , one is used to confirm the conjecture for all K 5 -minor free graphs, the other is devoted to proving that l a ( G ) = ⌈ Δ ( G ) 2 ⌉ holds if Δ ( G ) ≥ 9 . In addition, we prove here that if G is a K 5 − -minor free graph and Δ ( G ) ≥ 5 , then l a ( G ) = ⌈ Δ ( G ) 2 ⌉ .

Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik–Chervonenkis Dimension

Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in . In particular for any fixed , the class of -minor free graphs has distance VC-dimension at most . Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most , for any fixed , either computes the diameter or concludes that it is larger than in time , where only depends on and the notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on for any graph class that excludes a fixed graph as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the -minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of -nets, region decomposition, and other partition techniques.

Improved bound for improper colourings of graphs with no odd clique minor

AbstractStrengthening Hadwiger’s conjecture, Gerards and Seymour conjectured in 1995 that every graph with no odd $K_t$ -minor is properly $(t-1)$ -colourable. This is known as the Odd Hadwiger’s conjecture. We prove a relaxation of the above conjecture, namely we show that every graph with no odd $K_t$ -minor admits a vertex $(2t-2)$ -colouring such that all monochromatic components have size at most $\lceil \frac{1}{2}(t-2) \rceil$ . The bound on the number of colours is optimal up to a factor of $2$ , improves previous bounds for the same problem by Kawarabayashi (2008, Combin. Probab. Comput.17 815–821), Kang and Oum (2019, Combin. Probab. Comput.28 740–754), Liu and Wood (2021, arXiv preprint, arXiv:1905.09495), and strengthens a result by van den Heuvel and Wood (2018, J. Lond. Math. Soc.98 129–148), who showed that the above conclusion holds under the more restrictive assumption that the graph is $K_t$ -minor-free. In addition, the bound on the component-size in our result is much smaller than those of previous results, in which the dependency on $t$ was given by a function arising from the graph minor structure theorem of Robertson and Seymour. Our short proof combines the method by van den Heuvel and Wood for $K_t$ -minor-free graphs with some additional ideas, which make the extension to odd $K_t$ -minor-free graphs possible.

Immersion and clustered coloring

Hadwiger and Hajós conjectured that for every positive integer t, Kt+1-minor free graphs and Kt+1-topological minor free graphs are properly t-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Hajós' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for H-immersion free graphs, for any fixed graph H, up to a small additive absolute constant. Our result is tight for infinitely many graphs H.A key machinery developed in this paper is a lemma that reduces a clustering coloring problem on graphs to the one on the torsos of their tree-cut decomposition or tree-decomposition. A byproduct of this machinery is a unified proof of a result of Alon, Ding, Oporowski and Vertigan and a result of the author and Oum about clustered coloring graphs of bounded maximum degree in minor-closed families.

Disproof of a Conjecture by Woodall on the Choosability of $K_{s,t}$-Minor-Free Graphs

In 2001, in a survey article about list coloring, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable.
 In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\varepsilon>0$ and $C \ge 1$ there exists $N=N(\varepsilon,C) \in \mathbb{N}$ such that for all integers $s,t $ with $N \le s \le t \le Cs$ there exists a graph without a $K_{s,t}$-minor and list chromatic number greater than $(1-\varepsilon)(2s+t)$.

Packing topological minors half‐integrally

The packing problem and the covering problem are two of the most general questions in graph theory. The Erd\H{o}s-P\'{o}sa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent to the Erd\H{o}s-P\'{o}sa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erd\H{o}s-P\'{o}sa property holds for packing and covering $H$-topological minors, for any fixed graph $H$, which easily implies Thomas' conjecture. In fact, we prove an even stronger statement in which those topological minors are rooted at any choice of prescribed subsets of vertices. A number of results on $H$-topological minor free or $H$-minor free graphs have conclusions or requirements tied to properties of $H$. Classes of graphs that can half-integrally pack only a bounded number of $H$-topological minors or $H$-minors are more general topological minor-closed or minor-closed families whose minimal obstructions are more complicated than $H$. Our theorem provides a general machinery to extend those results to those more general classes of graphs without losing their tight connections to $H$.

A Note on Oct1+-Minor-Free Graphs and Oct2+-Minor-Free Graphs

Let [Formula: see text] and [Formula: see text] be the planar and non-planar graphs, respectively, obtained from the Octahedron by 3-splitting a vertex. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is [Formula: see text], [Formula: see text] [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices. We also show that a planar graph is [Formula: see text]-minor-free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from [Formula: see text], where [Formula: see text] is the set of graphs obtained by repeatedly taking the special 3-sums of [Formula: see text] and [Formula: see text] is the graph obtained from two 5-cycles [Formula: see text], [Formula: see text] by adding the five edges [Formula: see text] for all [Formula: see text]. For [Formula: see text], we prove that if a 4-connected graph is [Formula: see text]-minor-free, then it is planar, [Formula: see text] [Formula: see text], the line graph of [Formula: see text] or it is obtained from [Formula: see text] by repeatedly 4-splitting vertices.

Fault-tolerant distance labeling for planar graphs

In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph G such that from the labels of any three vertices u,v,f we can infer the u-to-v distance in the graph G∖{f}. We show that any directed weighted planar graph (and in fact any graph in a graph family with O(n)-size separators, such as minor-free graphs) admits fault-tolerant distance labels of size O(n2/3). We extend these labels in a way that allows us to also count the number of shortest paths, and provide additional upper and lower bounds for labels and oracles for counting shortest paths.

Coloring Drawings of Graphs

We consider cell colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only finitely many crossings, we define a cell $k$-coloring of $\Gamma(G)$ to be a coloring of the maximal connected regions of the drawing, the cells, with $k$ colors such that adjacent cells have different colors. By the $4$-color theorem, every drawing of a bridgeless graph has a cell $4$-coloring. A drawing of a graph is cell $2$-colorable if and only if the underlying graph is Eulerian. We show that every graph without degree 1 vertices admits a cell $3$-colorable drawing. This leads to the natural question which abstract graphs have the property that each of their drawings has a cell $3$-coloring. We say that such a graph is universally cell $3$-colorable. We show that every $4$-edge-connected graph and every graph admitting a nowhere-zero $3$-flow is universally cell $3$-colorable. We also discuss circumstances under which universal cell $3$-colorability guarantees the existence of a nowhere-zero $3$-flow. On the negative side, we present an infinite family of universally cell $3$-colorable graphs without a nowhere-zero $3-flow. On the positive side, we formulate a conjecture which has a surprising relation to a famous open problem by Tutte known as the $3$-flow-conjecture. We prove our conjecture for subcubic and for $K_{3,3}$-minor-free graphs.

Local Certification of Graph Decompositions and Applications to Minor-Free Classes

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 set of forbidden minors. In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor $H$, $H$-minor-free graphs can indeed be certified with labels of size $O(\log n)$. We also show matching lower bounds with a simple new proof technique.