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 + ϵ, from which we derive a (34 + ϵ)-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 + ϵ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces.

One of the most successful theoretical models in distributed computing is LOCAL, introduced in a seminal work by Linial [SIAM J. Comp. 1992]. Over the years, when studying distributed graph problems in the LOCAL model, researchers made different assumptions on the exact details of this model. For example, sometimes it is assumed that all machines know the exact size of the network, other times machines are assumed to only know a polynomial upper bound on the size of the network, while sometimes no prior knowledge is assumed. Are these small differences irrelevant details or do they actually heavily affect the obtained results? We investigate how robust our current understanding of the LOCAL model truly is, by focusing on one of the most studied classes of problems, called Locally Checkable Labelings (LCLs).

In property testing, a tester makes queries to (an oracle for) a graph and, on a graph having or being far from having a property P, it decides with high probability whether the graph satisfies P or not. Often, testers are restricted to a constant number of queries. While the graph properties for which there exists such a tester are somewhat well characterized in the dense graph model, it is not the case for sparse graphs. In this area, Czumaj and Sohler (FOCS’19) proved that H-freeness (i.e. the property of excluding the graph H as a subgraph) can be tested with constant queries on planar graphs as well as on graph classes excluding a minor. Using results from the sparsity toolkit, we propose a simpler alternative to the proof of Czumaj and Sohler, for a statement generalized to the broader notion of bounded expansion. That is, we prove that for any class 𝒞 with bounded expansion and any graph H, testing H-freeness can be done with constant query complexity on any graph G in 𝒞, where the constant depends on H and 𝒞, but is independent of G. While classes excluding a minor are prime examples of classes with bounded expansion, so are, for example, cubic graphs, graph classes with bounded maximum degree, or graphs of bounded book thickness. Additionally, random graphs with bounded average degree almost surely have bounded expansion.

Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one; namely, we show that every graph whose edge set can be covered by $k$ shortest paths has pathwidth $O(k^4)$, answering a question from the same paper. Moreover, we prove that when $k\leq 3$, every such graph has pathwidth at most $k$ (and this bound is tight). Finally, we show that even though there exist graphs with arbitrarily large treewidth whose vertex set can be covered by $2$ isometric trees, every graph whose set of edges can be covered by $2$ isometric trees has treewidth at most $2$.

We exhibit a new obstacle to the nascent algorithmic theory for classes excluding an induced minor. We indeed show that on the class of string graphs -- which avoids the 1-subdivision of, say, $K_5$ as an induced minor -- Induced 2-Disjoint Paths is NP-complete. So, while $k$-Disjoint Paths, for a fixed $k$, is polynomial-time solvable in general graphs, the absence of a graph as an induced minor does not make its induced variant tractable, even for $k=2$. This answers a question of Korhonen and Lokshtanov [SODA '24], and complements a polynomial-time algorithm for Induced $k$-Disjoint Paths in classes of bounded genus by Kobayashi and Kawarabayashi [SODA '09]. In addition to being string graphs, our produced hard instances are subgraphs of a constant power of bounded-degree planar graphs, hence have bounded twin-width and bounded maximum degree. We also leverage our new result to show that there is a fixed subcubic graph $H$ such that deciding if an input graph contains $H$ as an induced subdivision is NP-complete. Until now, all the graphs $H$ for which such a statement was known had a vertex of degree at least 4. This answers a question by Chudnovsky, Seymour, and the fourth author [JCTB '13], and by Le [JGT '19]. Finally we resolve another question of Korhonen and Lokshtanov by exhibiting a subcubic graph $H$ without two adjacent degree-3 vertices and such that deciding if an input $n$-vertex graph contains $H$ as an induced minor is NP-complete, and unless the Exponential-Time Hypothesis fails, requires time $2^{Ω(\sqrt n)}$. This complements an algorithm running in subexponential time $2^{O(n^{2/3} \log n)}$ by these authors [SODA '24] under the same technical condition.

We show that graphs excluding K2,t as a minor admit a f(t)-round 50-approximation deterministic distributed algorithm for Minimum Dominating Set. The result extends to Minimum Vertex Cover. Though fast and approximate distributed algorithms for such problems were already known for H-minor-free graphs, all of them have an approximation ratio depending on the size of H. To the best of our knowledge, this is the first example of a large non-trivial excluded minor leading to fast and constant-approximation distributed algorithms, where the ratio is independent of the size of H. A new key ingredient in the analysis of these distributed algorithms is the use of asymptotic dimension.

One of the cornerstones of the distributed complexity theory is the derandomization result by Chang, Kopelowitz, and Pettie [FOCS 2016]: any randomized LOCAL algorithm that solves a locally checkable labeling problem (LCL) can be derandomized with at most exponential overhead. The original proof assumes that the number of random bits is bounded by some function of the input size. We give a new, simple proof that does not make any such assumptions-it holds even if the randomized algorithm uses infinitely many bits. While at it, we also broaden the scope of the result so that it is directly applicable far beyond LCL problems.