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Nikolai Karol

Publications and source records attributed to Nikolai Karol.

7 recordsLinked to original sources
arXiv2026

Countable Graphs with Finite Path-width: Characterisation and Universality

We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.

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arXiv2026

Restriction on minimum degree in the contractible sets problem

Let $G$ be a $3$-connected graph. A set $W \subset V(G)$ is called contractible if $G(W)$ is a connected graph and $G - W$ is a $2$-connected graph. In 1994, McCuaig and Ota conjectured that for any $k \in \mathbb{N}$ there exists $n \in \mathbb{N}$ such that any 3-connected graph $G$ with $v(G) \geqslant n$ has a $k$-vertex contractible set. It is proved that this holds if $k \geqslant 5$ and $δ(G) \geqslant \left[ \frac{2k + 1}{3} \right] + 2$.

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arXiv2025

String Graphs: Product Structure and Localised Representations

We investigate string graphs through the lens of graph product structure theory, which describes complicated graphs as subgraphs of strong products of simpler building blocks. A graph $G$ is called a string graph if its vertices can be represented by a collection $\mathcal{C}$ of continuous curves (called a string representation of $G$) in a surface so that two vertices are adjacent in $G$ if and only if the corresponding curves in $\mathcal{C}$ cross. We prove that every string graph with bounded maximum degree in a fixed surface is isomorphic to a subgraph of the strong product of a graph with bounded treewidth and a path. This extends recent product structure theorems for string graphs. Applications of this result are presented. This product structure theorem ceases to be true if the `bounded maximum degree' assumption is relaxed to `bounded degeneracy'. For string graphs in the plane, we give an alternative proof of this result. Specifically, we show that every string graph in the plane has a `localised' string representation where the number of crossing points on the curve representing a vertex $u$ is bounded by a function of the degree of $u$. Our proof of the product structure theorem also leads to a result about the treewidth of outerstring graphs, which qualitatively extends a result of Fox and Pach [Eur. J. Comb. 2012] about outerstring graphs with bounded maximum degree. We extend our result to outerstring graphs defined in arbitrary surfaces.

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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

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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