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David R. Wood

Publications and source records attributed to David R. Wood.

At least 37 records · Page 2Linked to original sources
arXiv2026

On Universal Graphs for Trees and Tree-Like Graphs

Chung and Graham [J. London Math. Soc. 1983] claimed to prove that there exists an $n$-vertex graph $G$ with $ \frac{5}{2}n \log_2 n + O(n)$ edges that contains every $n$-vertex tree as a subgraph. Frati, Hoffmann and Tóth [Combin. Probab. Comput. 2023] discovered an error in the proof. By adding more edges to $G$ the error can be corrected, bringing the number of edges in $G$ to $\frac{7}{2}n \log_2 n + O(n). $ We make the first improvement to Chung and Graham's bound in over four decades by showing that there exists an $n$-vertex graph with $ \frac{14}{5}n \log_2 n + O(n) $ edges that contains every $n$-vertex tree as a subgraph. Furthermore, we generalise this bound for treewidth-$k$ graphs by showing that there exists a graph with $O(kn\log(n/k+1))$ edges that contains every $n$-vertex treewidth-$k$ graph as a subgraph. This is best possible in the sense that $Ω(kn\log(n/k+1))$ edges are required.

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arXiv2025

Verifying Hadwiger's Conjecture for Examples of Graphs with $α(G) = 2$

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most important open problems in graph theory. The case of Hadwiger's Conjecture for graphs with $α(G) = 2$ has garnered much attention. Seymour writes: ``My own belief is, if Hadwiger's Conjecture is true for graphs with stability number two then it is probably true in general, so it would be very nice to decide this case.'' This paper presents several tools useful for proving that a graph $G$ with $α(G) = 2$ satisfies Hadwiger's Conjecture. In doing so, we survey and generalise several classical results on the $α(G) = 2$ case of Hadwiger's Conjecture. Further, we apply these tools to prove variants of Hadwiger's Conjecture for several noteworthy classes of graphs with $α(G) = 2$. In particular, we prove Hadwiger's Conjecture for inflations of the complements of the following graphs: graphs with girth at least $5$, triangle-free Kneser graphs, and the Clebsch, Mesner, and Gewirtz graphs. This paper also highlights classes of graphs with $α(G) = 2$ where it is unknown if Hadwiger's Conjecture holds.

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arXiv2025

Quasi-tree-partitions of graphs with an excluded subgraph

This paper studies the structure of graphs with given tree-width and excluding a fixed complete bipartite subgraph, which generalises the bounded degree setting. We give a new structural description of such graphs in terms of so-called quasi-tree-partitions. We demonstrate the utility of this result through applications to (fractional) clustered colouring. Further generalisations of these structural and colouring results are presented.

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arXiv2025

Optimal Tree-Decompositions with Bags of Bounded Treewidth

We prove that several natural graph classes have tree-decompositions with minimum width such that each bag has bounded treewidth. For example, every planar graph has a tree-decomposition with minimum width such that each bag has treewidth at most 3. This treewidth bound is best possible. More generally, every graph of Euler genus $g$ has a tree-decomposition with minimum width such that each bag has treewidth in $O(g)$. This treewidth bound is best possible. Most generally, every $K_p$-minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function $f(p)$. In such results, the assumption of an excluded minor is justified, since we show that analogous results do not hold for the class of 1-planar graphs, which is one of the simplest non-minor-closed monotone classes. In fact, we show that 1-planar graphs do not have tree-decompositions with width within an additive constant of optimal, and with bags of bounded treewidth. On the other hand, we show that 1-planar $n$-vertex graphs have tree-decompositions with width $O(\sqrt{n})$ (which is the asymptotically tight bound) and with bounded treewidth bags. Moreover, this result holds in the more general setting of bounded layered treewidth, where the union of a bounded number of bags has bounded treewidth.

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arXiv2025

Defective and Clustered Colouring of Graphs with Given Girth

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$. Similarly, the clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $c$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has at most $c$ vertices. This paper determines or establishes bounds on the defective and clustered chromatic numbers of graphs with given girth in minor-closed classes defined by the following parameters: Hadwiger number, treewidth, pathwidth, treedepth, circumference, and feedback vertex number. One striking result is that for any integer $k$, for the class of triangle-free graphs with treewidth $k$, the defective chromatic number, clustered chromatic number and chromatic number are all equal. The same result holds for graphs with treedepth $k$, and generalises for graphs with no $K_p$ subgraph.

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arXiv2025

Reduced bandwidth: a qualitative strengthening of twin-width in minor-closed classes (and beyond)

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying $u$ and $v$, each edge incident to exactly one of $u$ and $v$ is coloured red. Bonnet, Kim, Thomassé and Watrigant [J. ACM 2022] defined the twin-width of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has maximum degree at most $k$. For any graph parameter $f$, we define the reduced $f$ of a graph $G$ to be the minimum integer $k$ such that there is a reduction sequence of $G$ in which every red graph has $f$ at most $k$. Our focus is on graph classes with bounded reduced bandwidth, which implies and is stronger than bounded twin-width (reduced maximum degree). We show that every proper minor-closed class has bounded reduced bandwidth, which is qualitatively stronger than an analogous result of Bonnet et al.\ for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least $2^{1000}$. We show that planar graphs have reduced bandwidth at most $466$ and twin-width at most $583$. Our bounds for graphs of Euler genus $γ$ are $O(γ)$. Lastly, we show that fixed powers of graphs in a proper minor-closed class have bounded reduced bandwidth (irrespective of the degree of the vertices). In particular, we show that map graphs of Euler genus $γ$ have reduced bandwidth $O(γ^4)$. Lastly, we separate twin-width and reduced bandwidth by showing that any infinite class of expanders excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth, while there are bounded-degree expanders with twin-width at most 6.

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arXiv2025

Expansion of gap-planar graphs

A graph is $k$-gap-planar if it has a drawing in the plane such that every crossing can be charged to one of the two edges involved so that at most $k$ crossings are charged to each edge. We show this class of graphs has linear expansion. In particular, every $r$-shallow minor of a $k$-gap-planar graph has density $O(rk)$. Several extensions of this result are proved: for topological minors, for $k$-cover-planar graphs, for $k$-gap-cover-planar graphs, and for drawings on any surface. Application to graph colouring are presented.

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arXiv2025

Structure of $k$-Matching-Planar Graphs

For $k \geqslant 0$, we define a simple topological graph $G$ (that is, a graph drawn in the plane such that every pair of edges intersect at most once, including endpoints) to be $k$-matching-planar if for every edge $e \in E(G)$, every matching amongst the edges of $G$ that cross $e$ has size at most $k$. The class of $k$-matching-planar graphs is a significant generalisation of many other existing beyond planar graph classes, including $k$-planar graphs. We prove that every simple topological $k$-matching-planar graph is isomorphic to a subgraph of the strong product of a graph with bounded treewidth and a path. This result qualitatively extends the planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] and recent product structure theorems for other beyond planar graph classes. Using this result, we deduce that the class of simple topological $k$-matching-planar graphs has several attractive properties, making it the broadest class of simple beyond planar graphs in the literature that has these properties. All of our results about simple topological $k$-matching-planar graphs generalise to the non-simple setting, where the maximum number of pairwise crossing edges incident to a common vertex becomes relevant. The paper introduces several tools and results of independent interest. We show that every simple topological $k$-matching-planar graph admits an edge-colouring with $\mathcal{O}(k^{3}\log k)$ colours such that monochromatic edges do not cross. As a key ingredient of the proof of our main product structure theorem, we introduce the concept of weak shallow minors, which subsume and generalise shallow minors, a key concept in graph sparsity theory. We also establish upper bounds on the treewidth of graphs with well-behaved circular drawings that qualitatively generalise several existing results.

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arXiv2025

Embedding Graphs of Simple Treewidth into Sparse Products

We study embeddings of graphs with bounded treewidth or bounded simple treewidth into the undirected graph underlying the directed product of two directed graphs. If the factors have bounded maximum indegrees, then the product graph has bounded maximum indegree and therefore is sparse. We prove that every graph of simple treewidth $k$ is contained in (ignoring edge directions) the directed product of directed graphs $\vec H_1$ and $\vec H_2$, with $Δ^-(\vec H_1), Δ^-(\vec H_2) \leq k-1 $ and $\text{tw}(H_1), \text{tw}(H_2) \leq k-1$. Further, we show that this treewidth bound is best possible. Several corollaries follow from our results: every outerplanar graph is contained in the directed product of trees with maximum indegree $1$, and every planar graph with treewidth $3$ is contained in a directed product of graphs with treewidth $2$ and maximum indegree $2$. However, for graphs of treewidth $k$, we prove a negative result: for any integers $s, t, k \geq 1$, there is a graph $G$ with treewidth $k$ not contained in the directed product of $\vec H_1$ and $\vec H_2$ for any directed graphs $\vec H_1$ and $\vec H_2$ with $Δ^-(\vec H_1) \leq s$, $Δ^-(\vec H_2) \leq t$, and $\text{tw}(H_1), \text{tw}(H_2) \leq k-1$. This result stands in stark contrast to the strong product case, where Liu, Norin and Wood [arXiv:2410.20333] proved that the optimal lower bound on the factors is about half the treewidth.

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arXiv2025

On universal graphs for trees and treewidth $k$ graphs

Let $s(n)$ be the minimum number of edges in a graph that contains every $n$-vertex tree as a subgraph. Chung and Graham [J. London Math. Soc. 1983] claim to prove that $s(n)\leqslant O(n\log n)$. We point out a mistake in their proof. The previously best known upper bound is $s(n)\leqslant O(n(\log n)(\log\log n)^{2})$ by Chung, Graham and Pippenger [Proc. Hungarian Coll. on Combinatorics 1976], the proof of which is missing many crucial details. We give a fully self-contained proof of the new and improved upper bound $s(n)\leqslant O(n(\log n)(\log\log n))$. The best known lower bound is $s(n)\geqslant Ω(n\log n)$. We generalise these results for graphs of treewidth $k$. For an integer $k\geqslant 1$, let $s_k(n)$ be the minimum number of edges in a graph that contains every $n$-vertex graph with treewidth $k$ as a subgraph. So $s(n)=s_1(n)$. We show that $Ω(k n\log n) \leqslant s_k(n) \leqslant O(kn(\log n)(\log\log n))$.

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arXiv2025

Grid Minors and Products

Motivated by recent developments regarding the product structure of planar graphs, we study relationships between treewidth, grid minors, and graph products. We show that the Cartesian product of any two connected $n$-vertex graphs contains an $Ω(\sqrt{n})\timesΩ(\sqrt{n})$ grid minor. This result is tight: The lexicographic product (which includes the Cartesian product as a subgraph) of a star and any $n$-vertex tree has no $ω(\sqrt{n})\timesω(\sqrt{n})$ grid minor.

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arXiv2025

A Graph-Theoretic Framework for Free-Parafermion Solvability

We present a graph-theoretic characterisation of when a quantum spin model admits an exact solution via a mapping to free parafermions. Our characterisation is based on the concept of a frustration graph, which represents the commutation relations between Weyl operators of a Hamiltonian. We show that a quantum spin system has an exact free-parafermion solution if its frustration graph is an oriented indifference graph. Further, we show that if the frustration graph of a model can be dipath oriented via switching operations, then the model is integrable in the sense that there is a family of commuting independent set charges. Additionally, we establish an efficient algorithm for deciding whether this is possible. Our characterisation extends that given for free-fermion solvability. Finally, we apply our results to solve three qudit spin models.

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arXiv2025

Treewidth 2 in the Planar Graph Product Structure Theorem

We prove that every planar graph is contained in $H_1\boxtimes H_2\boxtimes K_2$ for some graphs $H_1$ and $H_2$ both with treewidth 2. This resolves a question of Liu, Norin and Wood [arXiv:2410.20333]. We also show this result is best possible: for any $c \in \mathbb{N}$, there is a planar graph $G$ such that for any tree $T$ and graph $H$ with $\text{tw}(H) \leqslant 2$, $G$ is not contained in $H \boxtimes T \boxtimes K_c$.

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arXiv2025

Polynomial Bounds in the Apex Minor Theorem

A graph $A$ is "apex" if $A-z$ is planar for some vertex $z\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\mathcal{G}$, the graphs in $\mathcal{G}$ with bounded radius have bounded treewidth if and only if some apex graph is not in $\mathcal{G}$. In particular, for every apex graph $A$ and integer $r$, there is a minimum integer $g(A,r)$ such that every $A$-minor-free graph with radius $r$ has treewidth at most $g(A,r)$. We show that if $t=|V(A)|$ then $g(A,r)\in O^\ast(r^9t^{18})$ which is the first upper bound on $g(A,r)$ with polynomial dependence on both $r$ and $t$. More precisely, we show that every $A$-minor-free graph with radius $r$ has no $16rt^2 \times 16rt^2$ grid minor, which implies the first result via the Polynomial Grid Minor Theorem. A key example of an apex graph is the complete bipartite graph $K_{3,t}$, since $K_{3,t}$-minor-free graphs include and generalise graphs embeddable in any fixed surface. In this case, we prove that every $K_{3,t}$-minor-free graph with radius $r$ has no $4r(1+\sqrt{t})\times 4r(1+\sqrt{t})$ grid minor, which is tight up to a constant factor.

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arXiv2025

Short Paths in the Planar Graph Product Structure Theorem

The Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM '20] says that every planar graph $G$ is contained in $H\boxtimes P\boxtimes K_3$ for some planar graph $H$ with treewidth at most 3 and some path $P$. This result has been the key to solving several old open problems. Several people have asked whether the Planar Graph Product Structure Theorem can be proved with good upper bounds on the length of $P$. No $o(n)$ upper bound was previously known for $n$-vertex planar graphs. We answer this question in the affirmative, by proving that for any $ε\in (0,1)$ every $n$-vertex planar graph is contained in $H\boxtimes P\boxtimes K_{O(1/ε)}$, for some planar graph $H$ with treewidth 3 and for some path $P$ of length $O(\frac{1}εn^{(1+ε)/2})$. This bound is almost tight since there is a lower bound of $Ω(n^{1/2})$ for certain $n$-vertex planar graphs. In fact, we prove a stronger result with $P$ of length $O(\frac{1}ε\,\textrm{tw}(G)\,n^ε)$, which is tight up to the $O(\frac{1}ε\,n^ε)$ factor for every $n$-vertex planar graph $G$. Finally, taking $ε=\frac{1}{\log n}$, we show that every $n$-vertex planar graph $G$ is contained in $H\boxtimes P\boxtimes K_{O(\log n)}$ for some planar graph $H$ with treewidth at most 3 and some path $P$ of length $O(\textrm{tw}(G)\,\log n)$. This result is particularly attractive since the treewidth of the product $H\boxtimes P\boxtimes K_{O(\log n)}$ is within a $O(\log^2n)$ factor of the treewidth of $G$.

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arXiv2024

Product Structure and Tree-Decompositions

This paper explores the structure of graphs defined by an excluded minor or an excluded odd minor through the lens of graph products and tree-decompositions. We prove that every graph excluding a fixed odd minor is contained in the strong product of two graphs each with bounded treewidth. For graphs excluding a fixed minor, we strengthen the result by showing that every such graph is contained in the strong product of two digraphs with bounded indegree and with bounded treewidth. This result has the advantage that the product now has bounded degeneracy. In the setting of 3-term products, we show that every $K_t$-minor-free graph is contained in $H_1\boxtimes H_2 \boxtimes K_{c(t)}$ where $\text{tw}(H_i)\leq t-2$. This treewidth bound is close to tight: in any such result with $\text{tw}(H_i)$ bounded, both $H_1$ and $H_2$ can be forced to contain any graph of treewidth $t-5$, implying $\text{tw}(H_1)\geq t-5$ and $\text{tw}(H_2)\geq t-5$. Analogous lower and upper bounds are shown for any excluded minor, where the minimum possible bound on $\text{tw}(H_i)$ is tied to the treedepth of the excluded minor. Subgraphs of the product of two graphs with bounded treewidth have two tree-decompositions where any bag from the first decomposition intersects any bag from the second decomposition in a bounded number, $k$, of vertices, so called $k$-orthogonal tree-decompositions. We show that graphs excluding a fixed odd-minor have a tree-decomposition and a path-decomposition that are $O(1)$-orthogonal. This implies that such graphs have a tree-decomposition in which each bag has bounded pathwidth. This result is best possible in that `pathwidth' cannot be replaced by `bandwidth' or `treedepth'. Moreover, we characterize the minor-closed classes that have a tree-decomposition in which each bag has bounded bandwidth, or each bag has bounded treedepth.

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arXiv2024

Treewidth, Hadwiger Number, and Induced Minors

Treewidth and Hadwiger number are two of the most important parameters in structural graph theory. This paper studies graph classes in which large treewidth implies the existence of a large complete graph minor. To formalise this, we say that a graph class $\mathcal{G}$ is (tw,had)-bounded if there is a function $f$ (called the (tw,had)-bounding function) such that tw$(G)$ $\leq$ $f$(had$(G)$) for every graph $G \in \mathcal{G}$. We characterise (tw,had)-bounded graph classes as those that exclude some planar graph as an induced minor, and use this characterisation to show that every proper vertex-minor-closed class is (tw,had)-bounded. Furthermore, we demonstrate that any (tw,had)-bounded graph class has a (tw,had)-bounding function in O(had$(G)^9$polylog(had$(G)$)). Our bound comes from the bound for the Grid Minor Theorem given by Chuzhoy and Tan, and any quantitative improvement to their result will lead directly to an improvement to our result. More strongly, we conjecture that every (tw,had)-bounded graph class has a linear (tw,had)-bounding function. In support of this conjecture, we show that it holds for the class of outer-string graphs, and for a natural generalisation of outer-string graphs: intersection graphs of strings rooted at the boundary of a fixed surface. We also verify our conjecture for low-rank perturbations of circle graphs, which is an important step towards verifying it for all proper vertex-minor-closed classes.

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