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Alexandra Wesolek

Publications and source records attributed to Alexandra Wesolek.

At least 19 recordsLinked to original sources
arXiv2026

Meta-Theorems for Cuttable Distributed Problems

We prove that given any $\alpha$-approximation LOCAL algorithm for Minimum Dominating Set (MDS) on planar graphs, we can construct an $f(g)$-round $(3\alpha+1)$-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 $\alpha=11+\varepsilon$, 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.

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arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

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, Milani\v{c}, Munaro, \v{S}torgel, 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 $\alpha$-degeneracy.

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arXiv2025

Distributed Approximation Algorithms for Minimum Dominating Set in Locally Nice Graphs

We give a new, short proof that graphs embeddable in a given Euler genus-$g$ surface admit a simple $f(g)$-round $\alpha$-approximation distributed algorithm for Minimum Dominating Set (MDS), where the approximation ratio $\alpha \le 906$. Using tricks from Heydt et al. [European Journal of Combinatorics (2025)], we in fact derive that $\alpha \le 34 +\varepsilon$, 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. All our distributed algorithms work in the deterministic LOCAL model. They do not require any preliminary embedding of the graph and only rely on two things: a LOCAL algorithm for MDS on planar graphs with ``uniform'' approximation guarantees and the knowledge that graphs embeddable in bounded Euler genus surfaces have asymptotic dimension $2$. More generally, our algorithms work in any graph class of bounded asymptotic dimension where ``most vertices'' are locally in a graph class that admits a LOCAL algorithm for MDS with uniform approximation guarantees.

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arXiv2025

Faithful universal graphs for minor-closed classes

It was proved by Huynh, Mohar, \v{S}\'amal, 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 $2^{\Omega(n)}$ 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, Ir\v{s}i\v{c}, 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.

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arXiv2025

Local Constant Approximation for Dominating Set on Graphs Excluding Large Minors

We show that graphs excluding $K_{2,t}$ 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.

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arXiv2024

Subgraph-universal planar graphs for trees

We show that there exists an outerplanar graph on $O(n^{c})$ vertices for $c = \log_2(3+\sqrt{10}) \approx 2.623$ that contains every tree on $n$ vertices as a subgraph. This extends a result of Chung and Graham from 1983 who showed that there exist (non-planar) $n$-vertex graphs with $O(n \log n)$ edges that contain all trees on $n$ vertices as subgraphs and a result from Gol'dberg and Livshits from 1968 who showed that there exists a universal tree for $n$-vertex trees on $n^{O(\log(n))}$ vertices. Furthermore, we determine the number of vertices needed in the worst case for a planar graph to contain three given trees as subgraph to be on the order of $\frac{3}{2}n$, even if the three trees are caterpillars. This answers a question recently posed by Alecu et al. in 2024. Lastly, we investigate (outer)planar graphs containing all (outer)planar graphs as subgraph, determining exponential lower bounds in both cases. We also construct a planar graph on $n^{O(\log(n))}$ vertices containing all $n$-vertex outerplanar graphs as subgraphs.

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arXiv2024

On the local resilience of random geometric graphs with respect to connectivity and long cycles

Given an increasing graph property $\mathcal{P}$, a graph $G$ is $α$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-α)$-proportion of its neighbours, $H$ has property $\mathcal{P}$. We study the above notion of local resilience with $G$ being a random geometric graph $G_d(n,r)$ obtained by embedding $n$ vertices independently and uniformly at random in $[0,1]^d$, and connecting two vertices by an edge if the distance between them is at most $r$. First, we focus on connectivity. We show that, for every $\varepsilon>0$, for $r$ a constant factor above the sharp threshold for connectivity $r_c$ of $G_d(n,r)$, the random geometric graph is $(1/2-\varepsilon)$-resilient for the property of being $k$-connected, with $k$ of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small $\varepsilon>0$, connectivity is not born $(1/2-\varepsilon)$-resilient in $2$-dimensional random geometric graphs. Second, we study local resilience with respect to the property of containing long cycles. We show that, for $r$ a constant factor above $r_c$, $G_d(n,r)$ is $(1/2-\varepsilon)$-resilient with respect to containing cycles of all lengths between constant and $2n/3$. Proving $(1/2-\varepsilon)$-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that $G_d(n,r)$ is $α$-resilient with respect to Hamiltonicity for a fixed constant $α= α(d)<1/2$.

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arXiv2024

A note on highly connected $K_{2,\ell}$-minor free graphs

We show that every $3$-connected $K_{2,\ell}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\ell$. As a consequence, we show that every 3-connected $K_{2,\ell}$-minor free graph with minimum degree at least $5$ and no twins of degree $5$ has bounded size. Our proofs use Steiner trees and nested cuts; in particular, they do not rely on Ding's characterization of $K_{2,\ell}$-minor free graphs.

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arXiv2024

Sparse graphs with bounded induced cycle packing number have logarithmic treewidth

A graph is $\mathcal{O}_k$-free if it does not contain $k$ pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) $\mathcal{O}_k$-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of $\mathcal{O}_2$-free graphs without $K_{2,3}$ as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in $\mathcal{O}_k$-free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set, Minimum Coloring) can be solved in polynomial time in sparse $\mathcal{O}_k$-free graphs, and that deciding the $\mathcal{O}_k$-freeness of sparse graphs is polynomial time solvable.

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arXiv2024

Cops and Robber on Hyperbolic Manifolds

The Cops and Robber game on geodesic spaces is a pursuit-evasion game with discrete steps which captures the behavior of the game played on graphs, as well as that of continuous pursuit-evasion games. One of the outstanding open problems about the game on graphs is to determine which graphs embeddable in a surface of genus $g$ have largest cop number. It is known that the cop number of genus $g$ graphs is $O(g)$ and that there are examples whose cop number is $\tildeΩ(\sqrt{g}\,)$. The same phenomenon occurs when the game is played on geodesic surfaces. In this paper we obtain a surprising result about the game on a surface with constant curvature. It is shown that two cops have a strategy to come arbitrarily close to the robber, independently of the genus. We also discuss upper bounds on the number of cops needed to catch the robber. Our results generalize to higher-dimensional hyperbolic manifolds.

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arXiv2023

A Note on the 2-Colored Rectilinear Crossing Number of Random Point Sets in the Unit Square

Let $S$ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of $S$ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that $S$ defines a pair of crossing edges of the same color is equal to $1/4$. This is connected to a recent result of Aichholzer et al. [GD 2019] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halfed. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation $\frac{1}{2}-\frac{7}{50}$ of the total number of crossings.

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arXiv2023

Reconfiguration of plane trees in convex geometric graphs

A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most $2n-4$ between any pair of non-crossing spanning trees (where $n$ denotes the number of points). Hernando et al. proved that the length of a minimal flip sequence can be of length at least $\frac 32 n$. Two recent results of Aichholzer et al. and Bousquet et al. improved the Avis and Fukuda upper bound by proving that there always exists a flip sequence of length respectively at most $2n - \log n$ and $2n - \sqrt{n}$. We improve the upper bound by a linear factor for the first time in 25 years by proving that there always exists a flip sequence between any pair of non-crossing spanning trees $T_1,T_2$ of length at most $c n$ where $c \approx 1.95$. Our result is actually stronger since we prove that, for any two trees $T_1,T_2$, there exists a flip sequence from $T_1$ to $T_2$ of length at most $c |T_1 \setminus T_2|$. We also improve the best lower bound in terms of the symmetric difference by proving that there exists a pair of trees $T_1,T_2$ such that a minimal flip sequence has length $\frac 53 |T_1 \setminus T_2|$, improving the lower bound of Hernando et al. by considering the symmetric difference instead of the number of vertices. We generalize this lower bound construction to non-crossing flips (where we close the gap between upper and lower bounds) and rotations.

cs.CG↗
arXiv2023

On Minimizing the Energy of a Spherical Graph Representation

Graph representations are the generalization of geometric graph drawings from the plane to higher dimensions. A method introduced by Tutte to optimize properties of graph drawings is to minimize their energy. We explore this minimization for spherical graph representations, where the vertices lie on a unit sphere such that the origin is their barycentre. We present a primal and dual semidefinite program which can be used to find such a spherical graph representation minimizing the energy. We denote the optimal value of this program by $ρ(G)$ for a given graph $G$. The value turns out to be related to the second largest eigenvalue of the adjacency matrix of $G$, which we denote by $λ_2$. We show that for $G$ regular, $ρ(G) \leq \frac{λ_{2}}{2} \cdot v(G)$, and that equality holds if and only if the $λ_{2}$ eigenspace contains a spherical 1-design. Moreover, if $G$ is a random $d$-regular graph, $ρ(G)=\left(\sqrt{(d-1)} +o(1)\right)\cdot v(G)$, asymptotically almost surely.

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arXiv2023

Sharp threshold for embedding balanced spanning trees in random geometric graphs

A rooted tree is balanced if the degree of a vertex depends only on its distance to the root. In this paper we determine the sharp threshold for the appearance of a large family of balanced spanning trees in the random geometric graph $\mathcal{G}(n,r,d)$. In particular, we find the sharp threshold for balanced binary trees. More generally, we show that all sequences of balanced trees with uniformly bounded degrees and height tending to infinity appear above a sharp threshold, and none of these appears below the same value. Our results hold more generally for geometric graphs satisfying a mild condition on the distribution of their vertex set, and we provide a polynomial time algorithm to find such trees.

math.CO↗
arXiv2023

Maximum Independent Set when excluding an induced minor: $K_1 + tK_2$ and $tC_3 \uplus C_4$

Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class excluding a fixed planar graph $H$ as an induced minor, Maximum Independent Set can be solved in polynomial time, and show that this is indeed the case when $H$ is any planar complete bipartite graph, or the 5-vertex clique minus one edge, or minus two disjoint edges. A positive answer would constitute a far-reaching generalization of the state-of-the-art, when we currently do not know if a polynomial-time algorithm exists when $H$ is the 7-vertex path. Relaxing tractability to the existence of a quasipolynomial-time algorithm, we know substantially more. Indeed, quasipolynomial-time algorithms were recently obtained for the $t$-vertex cycle, $C_t$ [Gartland et al., STOC '21] and the disjoint union of $t$ triangles, $tC_3$ [Bonamy et al., SODA '23]. We give, for every integer $t$, a polynomial-time algorithm running in $n^{O(t^5)}$ when $H$ is the friendship graph $K_1 + tK_2$ ($t$ disjoint edges plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm running in $n^{O(t^2 \log n)+f(t)}$, with $f$ a single-exponential function, when $H$ is $tC_3 \uplus C_4$ (the disjoint union of $t$ triangles and a 4-vertex cycle). The former extends a classical result on graphs excluding $tK_2$ as an induced subgraph [Alekseev, DAM '07], while the latter extends Bonamy et al.'s result.

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arXiv2022

A tight local algorithm for the minimum dominating set problem in outerplanar graphs

We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a $(5-\varepsilon)$-approximation, for any $\varepsilon>0$. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.

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arXiv2022

Every graph is eventually Turán-good

Let $H$ be a graph. We show that if $r$ is large enough as a function of $H$, then the $r$-partite Turán graph maximizes the number of copies of $H$ among all $K_{r+1}$-free graphs on a given number of vertices. This confirms a conjecture of Gerbner and Palmer.

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