IA Scholar Query: Treewidth Lower Bounds with Brambles.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 03 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Scramble number and tree-cut decompositions
https://scholar.archive.org/work/jcgdiyox7vfuxhvxwdjmendiae
The scramble number of a graph is an invariant recently developed to study chip-firing games and divisorial gonality. In this paper we introduce the screewidth of a graph, based on a variation of the existing literature on tree-cut decompositions. We prove that this invariant serves as an upper bound on scramble number, though they are not always equal. We study properties of screewidth, and present results and conjectures on its connection to divisorial gonality.Lisa Cenek, Lizzie Ferguson, Eyobel Gebre, Cassandra Marcussen, Jason Meintjes, Ralph Morrison, Liz Ostermeyer, Shefali Ramakrishna, Ben Weberwork_jcgdiyox7vfuxhvxwdjmendiaeSat, 03 Sep 2022 00:00:00 GMTKilling a Vortex
https://scholar.archive.org/work/r5tvlz2axnb6hfulsufy2huzhm
The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every t∈ℕ, there exists some constant c_t such that every K_t-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most c_t vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most c_t and "at most c_t vortices of depth c_t". Our main combinatorial result is a "vortex-free" refinement of the above structural theorem as follows: we identify a (parameterized) graph H_t, called shallow vortex grid, and we prove that if in the above structural theorem we replace K_t by H_t, then the resulting decomposition becomes "vortex-free". Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some H_t, the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an H_t-minor-free graph G, computes the generating function of all perfect matchings of G in polynomial time. This algorithm yields, on H_t-minor-free graphs, polynomial algorithms for computational problems such as the dimer problem, the exact matching problem, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every H_t as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.Dimitrios M. Thilikos, Sebastian Wiederrechtwork_r5tvlz2axnb6hfulsufy2huzhmSun, 31 Jul 2022 00:00:00 GMTCombing a Linkage in an Annulus
https://scholar.archive.org/work/lqzppwwee5ayhpuzxnb6pdkqom
A linkage in a graph G of size k is a subgraph L of G whose connected components are k paths. The pattern of a linkage of size k is the set of k pairs formed by the endpoints of these paths. A consequence of the Unique Linkage Theorem is the following: there exists a function f:ℕ→ℕ such that if a plane graph G contains a sequence 𝒞 of at least f(k) nested cycles and a linkage of size at most k whose pattern vertices lay outside the outer cycle of 𝒞, then G contains a linkage with the same pattern avoiding the inner cycle of 𝒞. In this paper we prove the following variant of this result: Assume that all the cycles in 𝒞 are "orthogonally" traversed by a linkage P and L is a linkage whose pattern vertices may lay either outside the outer cycle or inside the inner cycle of 𝒞:=[C_1,...,C_p,...,C_2p-1]. We prove that there are two functions g,f:ℕ→ℕ, such that if L has size at most k, P has size at least f(k), and |𝒞|≥ g(k), then there is a linkage with the same pattern as L that is "internally combed" by P, in the sense that L∩ C_p⊆ P∩ C_p. In fact, we prove this result in the most general version where the linkage L is s-scattered: no two vertices of distinct paths of L are within distance at most s. We deduce several variants of this result in the cases where s=0 and s>0. These variants permit the application of the unique linkage theorem on several path routing problems on embedded graphs.Petr A. Golovach and Giannos Stamoulis and Dimitrios M. Thilikoswork_lqzppwwee5ayhpuzxnb6pdkqomMon, 11 Jul 2022 00:00:00 GMTBounds on higher graph gonality
https://scholar.archive.org/work/6vvh7qcaprb6bgrlrqcus4rkce
We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving independence number, and lower bounds on gonality by scramble number. We apply our bounds to study the computational complexity of computing higher gonalities, proving that it is NP-hard to compute the second gonality of a graph when restricting to multiplicity-free divisors.Lisa Cenek, Lizzie Ferguson, Eyobel Gebre, Cassandra Marcussen, Jason Meintjes, Ralph Morrison, Liz Ostermeyer, Shefali Ramakrishnawork_6vvh7qcaprb6bgrlrqcus4rkceTue, 14 Jun 2022 00:00:00 GMTMatching minors in bipartite graphs
https://scholar.archive.org/work/ulk6jjkjnzh2hnh5ex5nky27a4
In this thesis we adapt fundamental parts of the Graph Minors series of Robertson and Seymour for the study of matching minors and investigate a connection to the study of directed graphs. We develope matching theoretic to established results of graph minor theory: We characterise the existence of a cross over a conformal cycle by means of a topological property. Furthermore, we develope a theory for perfect matching width, a width parameter for graphs with perfect matchings introduced by Norin. here we show that the disjoint alternating paths problem can be solved in polynomial time on graphs of bounded width. Moreover, we show that every bipartite graph with high perfect matching width must contain a large grid as a matching minor. Finally, we prove an analogue of the we known Flat Wall theorem and provide a qualitative description of all bipartite graphs which exclude a fixed matching minor.Sebastian Wiederrecht, Technische Universität Berlin, Stephan Kreutzerwork_ulk6jjkjnzh2hnh5ex5nky27a4Tue, 19 Apr 2022 00:00:00 GMTConstant Congestion Brambles
https://scholar.archive.org/work/nu4pgwwfondrbdzh4ut7h2mdpe
A bramble in an undirected graph $G$ is a family of connected subgraphs of $G$ such that for every two subgraphs $H_1$ and $H_2$ in the bramble either $V(H_1) \cap V(H_2) \neq \emptyset$ or there is an edge of $G$ with one endpoint in $V(H_1)$ and the second endpoint in $V(H_2)$. The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph $G$ equals one plus the treewidth of $G$. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree $n$-vertex expander a bramble of order $\Omega(n^{1/2+\delta})$ requires size exponential in $\Omega(n^{2\delta})$ for any fixed $\delta \in (0,\frac{1}{2}]$. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth $k$ admits a bramble of order $\widetilde{\Omega}(k^{1/2})$ and size $\widetilde{\mathcal{O}}(k^{3/2})$. ($\widetilde{\Omega}$ and $\widetilde{\mathcal{O}}$ hide polylogarithmic factors and divisors, respectively.) In this note, we first sharpen the second bound by proving that every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetilde{\Omega}(k^{1/2})$ and congestion $2$, i.e., every vertex of $G$ is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every $\delta \in (0,\frac{1}{2}]$, every graph $G$ of treewidth at least $k$ contains a bramble of order $\widetilde{\Omega}(k^{1/2+\delta})$ and size $2^{\widetilde{\mathcal{O}}(k^{2\delta})}$.Meike Hatzel, Pawel Komosa, Marcin Pilipczuk, Manuel Sorgework_nu4pgwwfondrbdzh4ut7h2mdpeThu, 31 Mar 2022 00:00:00 GMTConstant Congestion Brambles
https://scholar.archive.org/work/73mxu6semzgt3pikpbye2bcoxq
A bramble in an undirected graph G is a family of connected subgraphs of G such that for every two subgraphs H_1 and H_2 in the bramble either V(H_1) ∩ V(H_2) ≠∅ or there is an edge of G with one endpoint in V(H_1) and the second endpoint in V(H_2). The order of the bramble is the minimum size of a vertex set that intersects all elements of a bramble. Brambles are objects dual to treewidth: As shown by Seymour and Thomas, the maximum order of a bramble in an undirected graph G equals one plus the treewidth of G. However, as shown by Grohe and Marx, brambles of high order may necessarily be of exponential size: In a constant-degree n-vertex expander a bramble of order Ω(n^1/2+δ) requires size exponential in Ω(n^2δ) for any fixed δ∈ (0,1/2]. On the other hand, the combination of results of Grohe and Marx and Chekuri and Chuzhoy shows that a graph of treewidth k admits a bramble of order Ω(k^1/2) and size 𝒪(k^3/2). (Ω and 𝒪 hide polylogarithmic factors and divisors, respectively.) In this note, we first sharpen the second bound by proving that every graph G of treewidth at least k contains a bramble of order Ω(k^1/2) and congestion 2, i.e., every vertex of G is contained in at most two elements of the bramble (thus the bramble is of size linear in its order). Second, we provide a tight upper bound for the lower bound of Grohe and Marx: For every δ∈ (0,1/2], every graph G of treewidth at least k contains a bramble of order Ω(k^1/2+δ) and size 2^𝒪(k^2δ).Meike Hatzel, Pawel Komosa, Marcin Pilipczuk, Manuel Sorgework_73mxu6semzgt3pikpbye2bcoxqWed, 30 Mar 2022 00:00:00 GMTLossy Planarization: A Constant-Factor Approximate Kernelization for Planar Vertex Deletion
https://scholar.archive.org/work/377llva5ajasfplsbob5tng6gi
In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the family F = K_5, K_3,3. Whenever the family F contains at least one planar graph, then F-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [Fomin, Lokshtanov, Misra, and Saurabh, FOCS'12]. The Vertex planarization problem is arguably the simplest setting for which F does not contain a planar graph and the existence of a constant-factor approximation or a polynomial kernelization remains a major open problem. In this work we show that Vertex planarization admits an algorithm which is a combination of both approaches. Namely, we present a polynomial A-approximate kernelization, for some constant A > 1, based on the framework of lossy kernelization [Lokshtanov, Panolan, Ramanujan, and Saurabh, STOC'17]. Simply speaking, when given a graph G and integer k, we show how to compute a graph G' on poly(k) vertices so that any B-approximate solution to G' can be lifted to an (A*B)-approximate solution to G, as long as A*B*OPT(G) <= k. In order to achieve this, we develop a framework for sparsification of planar graphs which approximately preserves all separators and near-separators between subsets of the given terminal set. Our result yields an improvement over the state-of-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time O(n^eps)-approximation algorithm, for any eps > 0, and a quasi-polynomial-time (log n)^O(1) approximation algorithm, both randomized [Kawarabayashi and Sidiropoulos, FOCS'17]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively O(OPT^eps) and (log OPT)^O(1).Bart M. P. Jansen, Michał Włodarczykwork_377llva5ajasfplsbob5tng6giFri, 04 Feb 2022 00:00:00 GMTOn Strict Brambles
https://scholar.archive.org/work/rzvecb3b3fg33hptti5jaabeai
A strict bramble of a graph G is a collection of pairwise-intersecting connected subgraphs of G. The order of a strict bramble B is the minimum size of a set of vertices intersecting all sets of B. The strict bramble number of G, denoted by sbn(G), is the maximum order of a strict bramble in G. The strict bramble number of G can be seen as a way to extend the notion of acyclicity, departing from the fact that (non-empty) acyclic graphs are exactly the graphs where every strict bramble has order one. We initiate the study of this graph parameter by providing three alternative definitions, each revealing different structural characteristics. The first is a min-max theorem asserting that sbn(G) is equal to the minimum k for which G is a minor of the lexicographic product of a tree and a clique on k vertices (also known as the lexicographic tree product number). The second characterization is in terms of a new variant of a tree decomposition called lenient tree decomposition. We prove that sbn(G) is equal to the minimum k for which there exists a lenient tree decomposition of G of width at most k. The third characterization is in terms of extremal graphs. For this, we define, for each k, the concept of a k-domino-tree and we prove that every edge-maximal graph of strict bramble number at most k is a k-domino-tree. We also identify three graphs that constitute the minor-obstruction set of the class of graphs with strict bramble number at most two. We complete our results by proving that, given some G and k, deciding whether sbn(G) ≤ k is an NP-complete problem.Emmanouil Lardas and Evangelos Protopapas and Dimitrios M. Thilikos and Dimitris Zoroswork_rzvecb3b3fg33hptti5jaabeaiSat, 15 Jan 2022 00:00:00 GMTOn the scramble number of graphs
https://scholar.archive.org/work/tuojypxatvgkxixp4fnz32tofe
The scramble number of a graph is an invariant recently developed to aid in the study of divisorial gonality. In this paper we prove that scramble number is NP-hard to compute, also providing a proof that computing gonality is NP-hard even for simple graphs, as well as for metric graphs. We also provide general lower bounds for the scramble number of a Cartesian product of graphs, and apply these to compute gonality for many new families of product graphs.Marino Echavarria, Max Everett, Robin Huang, Liza Jacoby, Ralph Morrison, Ben Weberwork_tuojypxatvgkxixp4fnz32tofeTue, 07 Dec 2021 00:00:00 GMTA Compound Logic for Modification Problems: Big Kingdoms Fall from Within
https://scholar.archive.org/work/lpjop6xt6zafjiqvz7ly5s4jvu
We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. We propose a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in monadic second-order logic and have models of bounded treewidth, while target sentences express first-order logic properties along with minor-exclusion. Our logic captures problems that are not definable in first order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modelling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or G-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model checking can be done in quadratic time. This algorithmic meta-theorem encompasses, unifies, and extends all known meta-algorithmic results on minor-closed graph classes. Moreover, all derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique.Fedor V. Fomin and Petr A. Golovach and Ignasi Sau and Giannos Stamoulis and Dimitrios M. Thilikoswork_lpjop6xt6zafjiqvz7ly5s4jvuThu, 04 Nov 2021 00:00:00 GMTA New Lower Bound on Graph Gonality
https://scholar.archive.org/work/iuw3nxdqgregxmotoqhaydgk64
We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone, but it is subgraph monotone and invariant under refinement. We compute the scramble number and gonality of several families of graphs for which these invariants are strictly greater than the treewidth.Michael Harp, Elijah Jackson, David Jensen, Noah Speeterwork_iuw3nxdqgregxmotoqhaydgk64Thu, 04 Nov 2021 00:00:00 GMTTwin-width I: tractable FO model checking
https://scholar.archive.org/work/mgq7n42vorbfxcr3w7at3twruu
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 by 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].Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, Rémi Watrigantwork_mgq7n42vorbfxcr3w7at3twruuMon, 25 Oct 2021 00:00:00 GMTA Flat Wall Theorem for Matching Minors in Bipartite Graphs
https://scholar.archive.org/work/b24xncufdngcrgbriy7y5fyghq
A major step in the graph minors theory of Robertson and Seymour is the transition from the Grid Theorem which, in some sense uniquely, describes areas of large treewidth within a graph, to a notion of local flatness of these areas in form of the existence of a large flat wall within any huge grid of an H-minor free graph. In this paper, we prove a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor. Our result builds on a a tight relationship between structural digraph theory and matching theory and allows us to deduce a Flat Wall Theorem for digraphs which substantially differs from a previously established directed variant of this theorem.Archontia C. Giannopoulou, Sebastian Wiederrechtwork_b24xncufdngcrgbriy7y5fyghqThu, 14 Oct 2021 00:00:00 GMTStructural Properties of Graph Products
https://scholar.archive.org/work/bpwppk2y4be5hk6p7x233qogsy
Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically studies various structural properties of cartesian, direct and strong products. In particular, we characterise when these graph products contain a given complete multipartite subgraph, determine tight bounds for their degeneracy, establish new lower bounds for the treewidth of cartesian and strong products, and characterise when they have bounded treewidth and when they have bounded pathwidth.Robert Hickingbotham, David R. Woodwork_bpwppk2y4be5hk6p7x233qogsySat, 02 Oct 2021 00:00:00 GMT