arXiv subjects

Rutger Campbell

Publications and source records attributed to Rutger Campbell.

At least 19 recordsLinked to original sources

arXiv2026

The excluded minors for $\mathsf{Z}_{3}$-gainable and regular biased graphs

We prove that a biased graph is gainable over the group $\mathsf{Z}_{3}$ if and only if it contains no minor isomorphic to $(4K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$. We develop a theory of "partial groups" that is analogous to that of partial fields, and we use this theory to show that a biased graph is gainable over every non-trivial group if and only if it is gainable over $\mathsf{Z}_{2}$ and $\mathsf{Z}_{3}$. From this we derive an independent proof of the theorem due to Gerards that a biased graph is gainable over every non-trivial group if and only if it has no minor isomorphic to $(3K_{2},\emptyset)$, $\pm K_{3}$, or $-K_{4}$.

math.CO↗

arXiv2026

The role of counting quantifiers in laminar set systems

Laminar set systems consist of non-crossing subsets of a universe with set inclusion essentially corresponding to the descendant relationship of a tree, the so-called laminar tree. Laminar set systems lie at the core of many graph decompositions such as modular decompositions, split decompositions, and bi-join decompositions. We show that from a laminar set system we can obtain the corresponding laminar tree by means of a monadic second order logic (MSO) transduction. This resolves an open question originally asked by Courcelle and is a satisfying resolution as MSO is the natural logic for set systems and is sufficient to define the property ``laminar''. Using results from Campbell et al. [STACS 2025], we can now obtain transductions for obtaining modular decompositions, co-trees, split decompositions and bi-join decompositions using MSO instead of CMSO. We further gain some insight into the expressive power of counting quantifiers and provide some results towards determining when counting quantifiers can be simulated in MSO in laminar set systems and when they cannot.

cs.LO↗

arXiv2026

CMSO-transducing tree-like graph decompositions

We give $\operatorname{CMSO}$-transductions that, given a graph $G$, output its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant $\operatorname{MSO}$, a strictly more expressive logic than $\operatorname{CMSO}$. Our methods more generally yield $\operatorname{C}_2 \operatorname{MSO}$-transductions that output the canonical decompositions of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

cs.LO↗

arXiv2026

Binary matroids and degree-boundedness for pivot-minors

We prove that for every bipartite graph $H$ and positive integer $s$, the class of $K_{s,s}$-subgraph-free graphs excluding $H$ as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every $K_{s,t}$-free bipartite circle graph with $s\le t$ has a vertex of degree at most $\max\{2s-2, t-1\}$ and provide examples showing that this is tight.

math.CO↗

arXiv2025

Blind cop-width and balanced minors of graphs

We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in the graph. The blind cop-width is the minimum number of cops needed to catch the robber on a given graph. We link it with other known graph parameters defined in terms of pursuit-evasion games, and show a new lower bound with respect to treewidth. The proof introduces the notion of balanced minors, where all branch sets of a minor model have equal size.

math.CO↗

arXiv2025

Turán's theorem for Dowling geometries

The Dowling geometry $Q_n(Γ)$, where $Γ$ is a finite group, is a matroid that generalizes the complete-graphic matroid $M(K_{n+1})$. We determine the maximum size of an $N$-free submatroid of $Q_n(Γ)$ for various choices of $N$, including subgeometries $Q_m(Γ')$, lines $U_{2,\ell}$, and graphic matroids $M(H)$. When the group $Γ$ is trivial and $N=M(K_t)$, this problem reduces to Turán's classical result in extremal graph theory. We show that when $Γ$ is nontrivial, a complex dependence on $Γ$ emerges, even when $N=M(K_4)$.

math.CO↗

arXiv2025

The excluded minors of the class of spike minors

Mayhew et al.\ (2021) posed the problem of showing that the minor-closed class of spikes and their minors has a finite set of excluded minors and describing all of them. In this paper, we resolve this problem.

math.CO↗

arXiv2025

Characterizing real-representable matroids with large average hyperplane-size

Generalizing a theorem of the first two authors and Geelen for planes, we show that, for a real-representable matroid $M$, either the average hyperplane-size in $M$ is at most a constant depending only on its rank, or each hyperplane of $M$ contains one of a set of at most $r(M)-2$ lines. Additionally, in the latter case, the ground set of $M$ has a partition $(E_{1}, E_{2})$, where $E_{1}$ can be covered by few flats of relatively low rank and $|E_{2}|$ is bounded. These results extend to complex-representable and orientable matroids. Finally, we formulate a high-dimensional generalization of a classic problem of Motzkin, Grünbaum, Erdős and Purdy on sets of red and blue points in the plane with no monochromatic blue line. We show that the solution to this problem gives a tight upper bound on $|E_{2}|$. We also discuss this high-dimensional problem in its own right, and prove some initial results.

math.CO↗

arXiv2025

The Erdős-Pósa property for circle graphs as vertex-minors

We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.

math.CO↗

arXiv2025

Metric graphs of negative type

The negative type inequalities of a metric space are closely tied to embeddability. A result by Gupta, Newman, and Rabinovich implies that if a metric graph $G$ does not contain a theta submetric as an embedding, then $G$ has negative type. We show the converse: if a metric graph $G$ contains a theta, then it does not have negative type.

math.CO↗

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.

math.CO↗

arXiv2024

Clustered Colouring of Graph Products

A colouring of a graph $G$ has clustering $k$ if the maximum number of vertices in a monochromatic component equals $k$. Motivated by recent results showing that many natural graph classes are subgraphs of the strong product of a graph with bounded treewidth and a path, this paper studies clustered colouring of strong products of two bounded treewidth graphs, where none, one, or both graphs have bounded degree. For example, in the case of two colours, if $n$ is the number of vertices in the product, then we show that clustering $Θ(n^{2/3})$ is best possible, even if one of the graphs is a path. However, if both graphs have bounded degree, then clustering $Θ(n^{1/2})$ is best possible. With three colours, if one of the graphs has bounded degree, then we show that clustering $Θ(n^{3/7})$ is best possible. However, if neither graph has bounded degree, then clustering $Ω(n^{1/2})$ is necessary. More general bounds for any given number of colours are also presented.

math.CO↗

arXiv2024

Optimal bounds for zero-sum cycles. I. Odd order

For a finite (not necessarily Abelian) group $(Γ,\cdot)$, let $n(Γ) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from $Γ$, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved $n(\mathbb{Z}_q)=O(q \log q)$. This was later improved to a linear bound of $n(Γ)\le 8|Γ|$ for every finite Abelian group by Mészáros and the last author, and then further to $n(Γ)\le 2|Γ|-1$ for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta. In this series of two papers we conclude this line of research by proving that $n(Γ)\le |Γ|+1$ for every finite group $(Γ,\cdot)$, which is the best possible such bound in terms of the group order and precisely determines the value of $n(Γ)$ for all cyclic groups as $n(\mathbb{Z}_q)=q+1$. In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.

math.CO↗

arXiv2024

Average plane-size in complex-representable matroids

Melchior's inequality implies that the average line-length in a simple, rank-$3$, real-representable matroid is less than $3$. A similar result holds for complex-representable matroids, using Hirzebruch's inequality, but with a weaker bound of $4$. We show that the average plane-size in a simple, rank-$4$, complex-representable matroid is bounded above by an absolute constant, unless the matroid is the direct-sum of two lines. We also prove that, for any integer $k$, in complex-representable matroids with rank at least $2k-1$, the average size of a rank-$k$ flat is bounded above by a constant depending only on $k$. Finally, we prove that, for any integer $r\ge 2$, the average flat-size in rank-$r$ complex-representable matroids is bounded above by a constant depending only on $r$. We obtain our results using a theorem, due to Ben Lund, that gives a good estimate on the number of rank-$k$ flats in a complex-representable matroid.

math.CO↗

arXiv2023

Decompositions into two linear forests of bounded lengths

For some $k \in \mathbb{Z}_{\geq 0}\cup \infty$, we call a linear forest $k$-bounded if each of its components has at most $k$ edges. We will say a $(k,\ell)$-bounded linear forest decomposition of a graph $G$ is a partition of $E(G)$ into the edge sets of two linear forests $F_k,F_\ell$ where $F_k$ is $k$-bounded and $F_\ell$ is $\ell$-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both $k$ and $\ell$ are at least $2$, NP-complete if $k\geq 9$ and $\ell =1$, and is in P for $(k,\ell)=(2,1)$. Before this, the only known NP-complete cases were the $(2,2)$ and $(3,3)$ cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than $3$-edge-colouring such graphs.

math.CO↗

arXiv2022

Graphs of Linear Growth have Bounded Treewidth

A graph class $\mathcal{G}$ has linear growth if, for each graph $G \in \mathcal{G}$ and every positive integer $r$, every subgraph of $G$ with radius at most $r$ contains $O(r)$ vertices. In this paper, we show that every graph class with linear growth has bounded treewidth.

math.CO↗

arXiv2022

Product structure of graph classes with bounded treewidth

We 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 ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. 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 ${t \geq \max\{s,3\}}$); 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.

math.CO↗

arXiv2019

On the complex-representable excluded minors for real-representability

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if $\mathbb{F}_1$-representability is not equivalent to $\mathbb{F}_2$-representability, then each $\mathbb{F}_1$-representable matroid is a minor of a $\mathbb{F}_2$-representable excluded minor for $\mathbb{F}_1$-representability.

math.CO↗