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Johannes Carmesin

Publications and source records attributed to Johannes Carmesin.

At least 19 recordsLinked to original sources
arXiv2026

A Graph Minors Approach to Temporal Sequences

We develop a structural approach to simultaneous embeddability in temporal sequences of graphs, inspired by graph minor theory. Our main result is a classification theorem for 2-connected temporal sequences: we identify five obstruction classes and show that every 2-connected temporal sequence is either simultaneously embeddable or admits a sequence of improvements leading to an obstruction. This structural insight leads to a polynomial-time algorithm for deciding the simultaneous embeddability of 2-connected temporal sequences. The restriction to 2-connected sequences is necessary, as the problem is NP-hard for connected graphs, while trivial for 3-connected graphs. As a consequence, our framework also resolves the rooted-tree SEFE problem, a natural extension of the well-studied sunflower SEFE. More broadly, our results demonstrate the applicability of graph minor techniques to evolving graph structures and provide a foundation for future algorithmic and structural investigations in temporal graph theory.

math.CO↗
arXiv2026

Stotting in positional games

We introduce variants of the Maker-Breaker and Waiter-Client games, which we call \emph{stotting}, in which a player grants a slight advantage to the opponent. We prove that a winning strategy in either stotting variant yields winning strategies for both Maker and Waiter in the classical setting. Several existing Maker strategies in the literature in fact win with stotting, and therefore automatically provide both classical winning strategies (and similarly for stotting Waiter). Knox previously disproved a conjecture of Beck asserting that whenever Maker wins the Maker-Breaker game, Waiter also wins the corresponding Waiter-Client game; in this sense, our framework may be viewed as a way of repairing Beck's conjecture.

math.CO↗
arXiv2025

Hardness of Planarity for Weak Temporal Sequences of 2-Connected Graphs

A weak deletion sequence is a sequence $(G_1,\ldots,G_n)$ of graphs so that for each $i\in[n-1]$ either $G_i$ is isomorphic to a subgraph of $G_{i+1}$, or vice versa: $G_{i+1}$ is isomorphic to a subgraph of $G_i$. We prove that determining the simultaneous planar embeddability of weak deletion sequences of $2$-connected graphs is NP-hard.

math.CO↗
arXiv2025

Canonical Decompositions of 3-Connected Graphs

We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or thickened $K_{3,m}$'s. Our construction is explicit, canonical, and has the following applications: we obtain a new theorem characterising all finite Cayley graphs as either essentially 4-connected, cycles, or complete graphs on at most four vertices, and we provide an automatic proof of Tutte's wheel theorem.

math.CO↗
arXiv2025

Characterising 4-tangles through a connectivity property

Every large $k$-connected graph-minor induces a $k$-tangle in its ambient graph. The converse holds for $k\le 3$, but fails for $k\ge 4$. This raises the question whether `$k$-connected' can be relaxed to obtain a characterisation of $k$-tangles through highly cohesive graph-minors. We show that this can be achieved for $k=4$ by proving that internally 4-connected graphs have unique 4-tangles, and that every graph with a 4-tangle $τ$ has an internally 4-connected minor whose unique 4-tangle lifts to $τ$.

math.CO↗
arXiv2025

How Local Separators Shape Community Structure in Large Networks

Community detection is a key tool for analyzing the structure of large networks. Standard methods, such as modularity optimization, focus on identifying densely connected groups but often overlook natural local separations in the graph. In this paper, we investigate local separator methods, which decompose networks based on structural bottlenecks rather than global connectivity. We systematically compare them with well-established community detection algorithms on large real-world networks. Our results show that local 1-separators consistently identify the densest communities, outperforming modularity-based methods in this regard, while local 2-separators reveal hierarchical structures but may over-fragment small clusters. These findings are particularly strong for road networks, suggesting practical applications in transportation and infrastructure analysis. Our study highlights local separators as a scalable and interpretable alternative for network decomposition.

cs.SI↗
arXiv2025

Towards a Stallings-type theorem for finite groups

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local covering: a usually infinite graph obtained by keeping all local structure of the original graph while unfolding all other structure as much as possible. We use local separators and local coverings to discover and prove a low-order Stallings-type result for finite nilpotent groups $Γ$: the $r$-local covering of some Cayley graph $G$ of $Γ$ has $\geq 2$ ends that are separated by $\leq 2$ vertices iff $G$ has an $r$-local separator of size $\leq 2$ and $Γ$ has order $>r$, iff $Γ$ is isomorphic to $C_i\times C_j$ for some $i>r$ and $j\in\{1,2\}$.

math.CO↗
arXiv2024

Local 2-separators

How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications include: 1. A unique decomposition theorem for graphs along their local 2-separators analogous to the 2-separator theorem; 2. an exact characterisation of graphs with no bounded subdivision of a wheel.

math.CO↗
arXiv2023

How to apply tree decomposition ideas in large networks?

Graph decompositions are the natural generalisation of tree decompositions where the decomposition tree is replaced by a genuine graph. Recently they found theoretical applications in the theory of sparsity, topological graph theory, structural graph theory and geometric group theory. We demonstrate applicability of graph decompositions on large networks by implementing an efficient algorithm and testing it on road networks.

cs.DM↗
arXiv2023

Embedding simply connected 2-complexes in 3-space

Firstly, we characterise the embeddability of simply connected locally 3-connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski's characterisation of graph planarity, by nine excluded minors. This answers questions of Lovász, Pardon and U. Wagner. The excluded minors are the cones over $K_5$ and $K_{3,3}$, five related constructions, and the remaining two are obtained from triangulations of the Möbius strip by attaching a disc at its central cycle. Secondly, we extend the above theorem to all simply connected 2-dimensional simplicial complexes.

math.CO↗
arXiv2023

Entanglements

Robertson and Seymour constructed for every graph $G$ a tree-decomposition that efficiently distinguishes all the tangles in $G$. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical. The key ingredient is an axiomatisation of 'local properties' of tangles. Generalisations to locally finite graphs and matroids are also discussed.

math.CO↗
arXiv2023

Outerspatial 2-complexes: Extending the class of outerplanar graphs to three dimensions

We introduce the class of outerspatial 2-complexes as the natural generalisation of the class of outerplanar graphs to three dimensions. Answering a question of O-joung Kwon, we prove that a locally 2-connected 2-complex is outerspatial if and only if it does not contain a surface of positive genus as a subcomplex and does not have a space minor that is a generalised cone over $K_4$ or $K_{2,3}$. This is applied to nested plane embeddings of graphs; that is, plane embeddings constrained by conditions placed on a set of cycles of the graph.

math.CO↗
arXiv2022

Dual matroids of 2-complexes -- revisited

We prove that simply connected local 2-dimensional simplicial complexes embed in 3-space if and only if their dual matroids are graphic. Examples are provided that the assumptions of simply connectedness and locality are necessary. This may be regarded as a 3-dimensional analogue of Whitney's planarity criterion from 1932.

math.CO↗
arXiv2022

On Andreae's Ubiquity Conjecture

A graph $H$ is ubiquitous if for every graph $G$ that for every natural number $n$ contains $n$ vertex-disjoint $H$-minors contains infinitely many vertex-disjoint $H$-minors. Andreae conjectured that every locally finite graph is ubiquitous. We give a disconnected counterexample to this conjecture. It remains open whether every connected locally finite graph is ubiquitous.

math.CO↗
arXiv2020

Every planar graph with the Liouville property is amenable

We introduce a strengthening of the notion of transience for planar maps in order to relax the standard condition of bounded degree appearing in various results, in particular, the existence of Dirichlet harmonic functions proved by Benjamini and Schramm. As a corollary we obtain that every planar non-amenable graph admits Dirichlet harmonic functions.

math.CO↗