arXiv subjects

Cyril Gavoille

Publications and source records attributed to Cyril Gavoille.

At least 19 recordsLinked to original sources

arXiv2026

Minor-Universal Graph for Graphs on Surfaces

We show that for integer every $n$ and every surface $Σ$, there is a graph embeddable on $Σ$ with at most $c n^2$ vertices that contains as a minor every $n$-vertex graph embeddable on $Σ$. The constant $c$ depends polynomially on the Euler genus of $Σ$. This generalizes a well-known result for planar graphs by Robertson, Seymour, and Thomas [Quickly Excluding a Planar Graph. J. Comb. Theory B, 1994], which states that the square grid on $4n^2$ vertices contains as a minor every $n$-vertex planar graph, an important step in showing that graphs excluding that graphs excluding a planar graph as a minor have bounded tree-width. According to Gorsky, Seweryn, and Wiederrecht [Polynomial Bounds for the Graph Minor Structure Theorem, FOCS '25], our construction provides the final key ingredient in the search for polynomial bounds in the decomposition of graphs excluding as minor a given $n$-vertex graph achieving tight bounds with respect to the Euler genus of the surface part.

cs.DM↗

arXiv2026

Adjacency labelling for proper minor-closed graph classes

We show that every proper minor-closed class of graphs admits a $(1+o(1))\log_2 n$-bit adjacency labelling scheme. Equivalently, for every proper minor-closed class $\mathcal{G}$ and every positive integer $n$ there exists an $n^{1+o(1)}$-vertex graph $U$ such that every $n$-vertex graph in $\mathcal{G}$ is isomorphic to an induced subgraph of $U$. Both results are optimal up to the lower order term. They generalize the corresponding results for planar graphs and apex-minor-free classes (Dujmović et al., J.~ACM 2021) to all proper minor-closed classes, answering the open question raised in that paper and anticipated earlier by Bonamy, Gavoille, and Pilipczuk (SODA 2020).

cs.DM↗

arXiv2026

Freeze-Tag with Return

In the standard Freeze-Tag Problem (FTP), an initially awake robot (the source) is in charge of waking up a swarm of sleeping robots by moving towards them, given that all the awake robots can participate in the awakening process. The goal is to minimize the makespan to wake up all robots assuming they move at unit speed. In this paper we introduce the Freeze-Tag-with-Return Problem (FTRP) variant, where the robots must eventually return to their initial positions. In the Euclidean plane with $n$ sleeping robots lying on the unit disk centered at the initial position of the source, we show a non-trivial relationship between FTP and FTRP by proving that the difference between the optimal makespan of both problems never exceeds $1.959$, and is at least $1.732$ in the worst-case. We also present several upper and lower bounds on the optimal makespan. In particular, we show that if the sleeping robots are in convex positions, then the optimal makespan is at most $2 + 2\sqrt{2}$, which is achieved by some instances. From an algorithmic point-of-view, we present single-exponential algorithms for general distance functions. In metric spaces, these algorithms are asymptotically optimal under the ETH, which we show via an NP-hardness reduction on unweighted graphs.

cs.DS↗

arXiv2026

Entanglement from Expansion: High Rank-Width in Deterministic Graphs

Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on $n$ vertices with a provably maximum rank-width $Θ(n)$. Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than $Θ(\sqrt{n})$.

cs.DM↗

arXiv2026

Meta-Theorems for Cuttable Distributed Problems

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.

cs.DC↗

arXiv2025

Lower Bounds for Induced-Universal Graphs

We give a series of new lower bounds on the minimum number of vertices required by a graph to contain every graph of a given family as induced subgraph. In particular, we show that this induced-universal graph for $n$-vertex planar graphs must have at least $10.52n$ vertices. We also show that the number of conflicting graphs to consider in order to beat this lower bound is at least $137$. In other words, any family of less than $137$ planar graphs of $n$ vertices has an induced-universal graph with less than $10.52n$ vertices, stressing the difficulty in beating such lower bounds. Similar results are developed for other graph families, including but not limited to, trees, outerplanar graphs, series-parallel graphs, $K_{3,3}$-minor free graphs. As a byproduct, we show that any family of $t$ graphs of $n$ vertices having small chromatic number and sublinear pathwidth, like any proper minor-closed family, has an induced-universal graph with less than $\frac{15}{7} \sqrt{t} \cdot n$ vertices. This is achieved by making a bridge between equitable colorings, combinatorial designs, and path-decompositions.

math.CO↗

arXiv2025

An Improved Bound for Plane Covering Paths

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $π(n)$ segments. We prove that $π(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.

cs.CG↗

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 $α$-approximation distributed algorithm for Minimum Dominating Set (MDS), where the approximation ratio $α\le 906$. Using tricks from Heydt et al. [European Journal of Combinatorics (2025)], we in fact derive that $α\le 34 +\varepsilon$, 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.

cs.DC↗

arXiv2025

Isometric-Universal Graphs for Trees

We consider the problem of finding the smallest graph that contains two input trees each with at most $n$ vertices preserving their distances. In other words, we look for an isometric-universal graph with the minimum number of vertices for two given trees. We prove that this problem can be solved in time $O(n^{5/2}\log{n})$. We extend this result to forests instead of trees, and propose an algorithm with running time $O(n^{7/2}\log{n})$. As a key ingredient, we show that a smallest isometric-universal graph of two trees essentially is a tree. Furthermore, we prove that these results cannot be extended. Firstly, we show that deciding whether there exists an isometric-universal graph with $t$ vertices for three forests is NP-complete. Secondly, we show that any smallest isometric-universal graph cannot be a tree for some families of three trees. This latter result has implications for greedy strategies solving the smallest isometric-universal graph problem.

cs.DS↗

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.

cs.DC↗

arXiv2025

Distributed Freeze Tag: a Sustainable Solution to Discover and Wake-up a Robot Swarm

The Freeze Tag Problem consists in waking up a swarm of robots starting with one initially awake robot. Whereas there is a wide literature of the centralized setting, where the location of the robots is known in advance, we focus in the distributed version where the location of the robots $¶$ are unknown, and where awake robots only detect other robots up to distance~$1$. Assuming that moving at distance $δ$ takes a time $δ$, we show that waking up of the whole swarm takes $O(ρ+\ell^2\log( ρ/\ell))$, where $ρ$ stands for the largest distance from the initial robot to any point of $¶$, and the $\ell$ is the connectivity threshold of $¶$. Moreover, the result is complemented by a matching lower bound in both parameters $ρ$ and $\ell$. We also provide other distributed algorithms, complemented with lower bounds, whenever each robot has a bounded amount of energy.

cs.DS↗

arXiv2024

Freeze-Tag in $L_1$ has Wake-up Time Five

The Freeze-Tag Problem, introduced in Arkin et al. (SODA'02) consists of waking up a swarm of $n$ robots, starting from a single active robot. In the basic geometric version, every robot is given coordinates in the plane. As soon as a robot is awakened, it can move towards inactive robots to wake them up. The goal is to minimize the wake-up time of the last robot, the makespan. Despite significant progress on the computational complexity of this problem and on approximation algorithms, the characterization of exact bounds on the makespan remains one of the main open questions. In this paper, we settle this question for the $\ell_1$-norm, showing that a makespan of at most $5r$ can always be achieved, where $r$ is the maximum distance between the initial active robot and any sleeping robot. Moreover, a schedule achieving a makespan of at most $5r$ can be computed in optimal time $O(n)$. Both bounds, the time and the makespan are optimal. This implies a new upper bound of $5\sqrt{2}r \approx 7.07r$ on the makespan in the $\ell_2$-norm, improving the best known bound so far $(5+2\sqrt{2}+\sqrt{5})r \approx 10.06r$.

cs.DS↗

arXiv2021

Isometric universal graphs

A subgraph $H$ of a graph $G$ is isometric if the distances between vertices in $H$ coincide with the distances between the corresponding vertices in $G$. We show that for any integer $n\ge 1$, there is a graph on $3^{n+O(\log^2 n)}$ vertices that contains isometric copies of all $n$-vertex graphs. Our main tool is a new type of distance labelling scheme, whose study might be of independent interest.

math.CO↗

arXiv2021

Adjacency Labelling for Planar Graphs (and Beyond)

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine if $uv$ is an edge of $G$. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every $n$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that every $n$-vertex planar graph is an induced subgraph of $U_n$. These results generalize to bounded genus graphs, apex-minor-free graphs, bounded-degree graphs from minor closed families, and $k$-planar graphs.

cs.DS↗

arXiv2021

Cops, Robbers, and Threatening Skeletons: Padded Decomposition for Minor-Free Graphs

We prove that any graph excluding $K_r$ as a minor has can be partitioned into clusters of diameter at most $Δ$ while removing at most $O(r/Δ)$ fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove $O(r^2/Δ)$ fraction of the edges. Our result is obtained by a new approach to relate the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops-and-robbers game on excluded-minor graphs can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter $O(r)$ and strong-diameter partitions with padding parameter $O(r^2)$ for $K_r$-free graphs, padding $O(k)$ for graphs with treewidth $k$, and padding $O(\log g)$ for graphs with genus $g$.

cs.DS↗

arXiv2020

Shorter Labeling Schemes for Planar Graphs

An \emph{adjacency labeling scheme} for a given class of graphs is an algorithm that for every graph $G$ from the class, assigns bit strings (labels) to vertices of $G$ so that for any two vertices $u,v$, whether $u$ and $v$ are adjacent can be determined by a fixed procedure that examines only their labels. It is known that planar graphs with $n$ vertices admit a labeling scheme with labels of bit length $(2+o(1))\log{n}$. In this work we improve this bound by designing a labeling scheme with labels of bit length $(\frac{4}{3}+o(1))\log{n}$. In graph-theoretical terms, this implies an explicit construction of a graph on $n^{4/3+o(1)}$ vertices that contains all planar graphs on $n$ vertices as induced subgraphs, improving the previous best upper bound of $n^{2+o(1)}$. Our scheme generalizes to graphs of bounded Euler genus with the same label length up to a second-order term. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time.

cs.DS↗

arXiv2017

A Fast Network-Decomposition Algorithm and its Applications to Constant-Time Distributed Computation

A partition $(C_1,C_2,...,C_q)$ of $G = (V,E)$ into clusters of strong (respectively, weak) diameter $d$, such that the supergraph obtained by contracting each $C_i$ is $\ell$-colorable is called a strong (resp., weak) $(d, \ell)$-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong $(exp\{O(\sqrt{ \log n \log \log n})\}, exp\{O(\sqrt{ \log n \log \log n})\})$-network-decompositions can be computed in distributed deterministic time $exp\{O(\sqrt{ \log n \log \log n})\}$. The result of Awerbuch et al. was improved by Panconesi and Srinivasan in 1992: in the latter result $d = \ell = exp\{O(\sqrt{\log n})\}$, and the running time is $exp\{O(\sqrt{\log n})\}$ as well. Much more recently Barenboim (2012) devised a distributed randomized constant-time algorithm for computing strong network decompositions with $d = O(1)$. However, the parameter $\ell$ in his result is $O(n^{1/2 + ε})$. In this paper we drastically improve the result of Barenboim and devise a distributed randomized constant-time algorithm for computing strong $(O(1), O(n^ε))$-network-decompositions. As a corollary we derive a constant-time randomized $O(n^ε)$-approximation algorithm for the distributed minimum coloring problem, improving the previously best-known $O(n^{1/2 + ε})$ approximation guarantee. We also derive other improved distributed algorithms for a variety of problems. Most notably, for the extremely well-studied distributed minimum dominating set problem currently there is no known deterministic polylogarithmic-time algorithm. We devise a {deterministic} polylogarithmic-time approximation algorithm for this problem, addressing an open problem of Lenzen and Wattenhofer (2010).

cs.DC↗

arXiv2017

Towards Plane Spanners of Degree 3

Let $S$ be a finite set of points in the plane that are in convex position. We present an algorithm that constructs a plane $\frac{3+4π}{3}$-spanner of $S$ whose vertex degree is at most 3. Let $Λ$ be the vertex set of a finite non-uniform rectangular lattice in the plane. We present an algorithm that constructs a plane $3\sqrt{2}$-spanner for $Λ$ whose vertex degree is at most 3. For points that are in the plane and in general position, we show how to compute plane degree-3 spanners with a linear number of Steiner points.

cs.CG↗