Maximilian Gorsky

Dario Cavallaro, Maximilian Gorsky, Stephan Kreutzer et al.

The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bounds behind these results have traditionally been so large that the resulting algorithms are regarded as "galactic". Our main result is a general irrelevant-vertex theorem for a common generalisation of $k$-Disjoint Paths and Rooted Minor Checking for graphs of size at most $d,$ commonly called the $(k,d)$-Folio problem. Specifically, we show that for any graph $G$ in which the $k$ terminals are chosen from some set $R,$ if the treewidth of $G$ exceeds $β(k,b,d)\in$ $2^{{\bf poly}(b + d)}$ $\cdot {\bf poly}(k)$ then we can locate an irrelevant vertex for the $(k,d)$-Folio problem. Here, the quantity $b$ is the bidimensionality of $R,$ that is, the largest $b$ for which a $(b\times b)$-grid minor in $G$ can be rooted on $R$. Thus, the exponential component of the irrelevant-vertex threshold is driven by the bound on the bidimensionality, rather than by the number of terminals, and we argue that this dependence is essentially optimal up to polynomial factors. As a consequence, the Linkage Function satisfies $\ell(k) \in 2^{{\bf poly}(k)}$. Beyond its structural significance, our result yields improved parameter dependencies for algorithms for Disjoint Paths and Rooted Minor Checking}, and provides a quantitative improvement for a broad range of graph-minor-based algorithmic frameworks.

J. Pascal Gollin, Maximilian Gorsky, Meike Hatzel et al.

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $ψ$ on a surface $Σ$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(ψ)$ of $ψ$, and let $R(G,ψ)$ be the radial graph of $(G, ψ)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(ψ)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $ψ$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(ψ) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,ψ)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.

Maximilian Gorsky, Michał T. Seweryn, Sebastian Wiederrecht

We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the $13$th entry of Robertson and Seymour's Graph Minors Series [JCTB 1990] and has since become a cornerstone in the algorithmic theory of graph minors. A wall where each brick is assigned a set of colours is said to be homogeneous if each brick is assigned the same set of colours. The Homogeneous Wall Lemma says that there exists a function $h$ that, given non-negative integers $q$ and $k$ and an $h(q,k)$-wall $W$ where each brick is assigned a, possibly empty, subset of $\{ 1, \ldots , q \}$ contains a $k$-wall $W'$ as a subgraph such that, if one assigns to each brick $B$ of $W'$ the union of the sets assigned to the bricks of $W$ in its interior, then $W'$ is homogeneous. It is well-known that $h(q,k) \in k^{\mathcal{O}(q)}$. The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of $h$ on $q$ usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that $h(q,k) \in \mathcal{O}(q^4 \cdot k^6)$, we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].