Alexandra Wesolek

Paul Bastide, Louis Esperet, Carla Groenland et al.

It was proved by Huynh, Mohar, Šámal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for fixed $t$, if a graph $G$ is $K_t$-minor-free and contains every $n$-vertex planar graph as a subgraph, then $G$ has $2^{Ω(n)}$ vertices. On the other hand, we construct a polynomial size $K_4$-minor-free graph containing every $n$-vertex tree as an induced subgraph, and a polynomial size $K_7$-minor-free graph containing every $n$-vertex $K_4$-minor-free graph as induced subgraph. This answers several problems raised recently by Bergold, Iršič, Lauff, Orthaber, Scheucher and Wesolek. We study more generally the order of universal graphs for various classes (of graphs of bounded degree, treedepth, pathwidth, or treewidth), if the universal graphs retain some of the structure of the original class.

Marthe Bonamy, Avinandan Das, Cyril Gavoille et al.

We prove that given any $α$-approximation LOCAL algorithm for Minimum Dominating Set (MDS) on planar graphs, we can construct an $f(g)$-round $(3α+1)$-approximation LOCAL algorithm for MDS on graphs embeddable in a given Euler genus-$g$ surface. Heydt et al. [European Journal of Combinatorics (2025)] gave an algorithm with $α=11+\varepsilon$, from which we derive a $(34 +\varepsilon)$-approximation algorithm for graphs of genus $g$, therefore improving upon the current state of the art of $24g+O(1)$ due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of $91+\varepsilon$ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces. We generalize this result into two directions: (1) by considering other graph problems studied in Distributed Computing such as Minimum $k$-Tuple Dominating Set, for which constant-round approximation algorithms were known for planar graphs, but not for graphs of bounded genus; and (2) by considering graph classes beyond bounded genus graphs, called locally nice, and relying on the asymptotic dimension of the class. We prove these results by a series of meta-theorems about cuttable minimization problems with constant-round approximation LOCAL algorithms. Roughly speaking, in cuttable problems, one can systematically extract small subgraphs whose solutions are in proportion to the global solution restricted to the neighbourhood of the subgraph.

Václav Blažej, J. Pascal Gollin, Tomáš Hons et al.

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, every $\{P_t,K_{\ell,\ell}\}$-free graph has bounded tree-independence number. We prove this conjecture for $t=5$ by showing that every $\{P_5,K_{\ell,\ell}\}$-free graph has tree-independence number at most $4\ell$. We also obtain related bounds for the weaker parameter of $α$-degeneracy.