Michał Pilipczuk

Édouard Bonnet, Hung Le, Marcin Pilipczuk et al.

Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.

Václav Blažej, Michał Pilipczuk, Evangelos Protopapas

Menger's Theorem is a fundamental result in graph theory. It states that if in a graph $G$ with distinguished sets of terminal vertices $S$ and $T$ there are no $k$ pairwise vertex-disjoint $S$-$T$ paths, then there is a set of less than $k$ vertices that intersects every $S$-$T$ path. In this work, we give a coarse variant of this result for planar and bounded genus graphs. Precisely, we prove that for every surface $Σ$ there is a function $f\colon \mathbb{N}\times \mathbb{N}\to \mathbb{N}$ such that for every pair of integers $d,k\in \mathbb{N}$ and a $Σ$-embeddable graph $G$ with distinguished sets of terminal vertices $S$ and $T$, if $G$ does not contain a family of $k$ $S$-$T$ paths that are pairwise at distance larger than $d$, then there is a set $X$ consisting of at most $f(d,k)$ vertices of $G$ such that every $S$-$T$ path is at distance at most $d$ from a vertex of $X$. This partially answers questions of Nguyen, Scott, and Seymour [arXiv:2508.14332], who proved that such a result cannot hold in general graphs. A key ingredient of our proof is a structure theorem from the developing ''colorful'' graph minor theory, where the focus is on studying the structure in a graph relative to some fixed subsets of annotated vertices. In our case, these annotated vertices are $S$ and $T$.

Daniel Dadush, Michał Pilipczuk, Amadeus Reinald et al.

Fix a parameter $k\in \mathbf{N}$. We give dynamic data structures that for a fully dynamic undirected graph $G$, updated over time by edge insertions and edge deletions, can answer the following queries: - Long $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $k$? - Long $(u,v)$-detour: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of length at least $\text{dist}_G(u,v)+k$? - Even/odd $(u,v)$-path: Given $u,v\in V(G)$, is there a path from $u$ to $v$ of even/odd length? The amortized time of executing an update or answering a query is $2^{O(k^3)} \log n + O(\log^2 n \log^2 \log n)$ in the first two cases, and $O(\log^2 n \log^2 \log n)$ in the last, where $n$ is the number of vertices of $G$. The first result is in sharp contrast with known conditional lower bounds for reporting paths of length at most $k$. Specifically, there is no data structure supporting queries about $(u,v)$-paths of length at most two in time $n^{o(1)}$ unless the Triangle Conjecture fails. Our main technical contribution is a mechanism of "delayed edge insertion" that works locally on the level of biconnected components.