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.

Abstract. Given a graph [Formula: see text] and an integer [Formula: see text], let [Formula: see text] denote the minimum number of colors required to color the edges of [Formula: see text] such that, in each color class, the subgraph induced by the edges of that color has all nonzero degrees congruent to 1 modulo [Formula: see text]. In 1992, Pyber proved that [Formula: see text] for every graph [Formula: see text], and posed the question of whether [Formula: see text] can be bounded solely in terms of [Formula: see text] for every [Formula: see text]. This question was answered in 1997 by Scott, who showed that [Formula: see text], and further asked whether [Formula: see text]. Recently, Botler, Colucci, and Kohayakawa (2023) answered Scott’s question affirmatively proving that [Formula: see text], and conjectured that the multiplicative constant could be reduced to 1. A step towards this latter conjecture was made in 2024 by Nweit and Yang, who improved the bound to [Formula: see text]. In this paper, we further improve the multiplicative constant to 9. More specifically, we prove that there is a function [Formula: see text] for which [Formula: see text] if [Formula: see text] is odd, and [Formula: see text] if [Formula: see text] is even. In doing so, we prove that [Formula: see text] for every [Formula: see text]-degenerate graph [Formula: see text], which plays a central role in our proof.

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.

A graph is Ok-free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We show that MAXIMUM INDEPENDENT SET and 3-COLORING in Ok-free graphs can be solved in quasi-polynomial time. As a main technical result, we establish that “sparse” (here, not containing large complete bipartite graphs as subgraphs) Ok-free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is proven sharp as there is an infinite family of O2-free graphs without K3,3-subgraph and whose treewidth is (at least) logarithmic. 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 Ok-free graphs, and that deciding the Ok-freeness of sparse graphs is polynomial time solvable. * This work was supported by the ANR projects DISTANCIA (ANR-17-CE40-0015), DIGRAPHS (ANR-19-CE48-0013-01), and TWIN-WIDTH (ANR-21-CE48-0014-01), by the LabEx PERSYVAL-lab (ANR-11-LABX-0025), and by the Vanier Canada Graduate Scholarships program. † The full version of the paper can be accessed at https://arxiv.org/abs/2206.00594

We construct asymptotically optimal adjacency labelling schemes for every hereditary class containing 2Ω(n2) n-vertex graphs as n→ ∞. This regime contains many classes of interest, for instance perfect graphs or comparability graphs, for which we obtain an adjacency labelling scheme with labels of n/4+o(n) bits per vertex. This implies the existence of a reachability labelling scheme for digraphs with labels of n/4+o(n) bits per vertex and comparability labelling scheme for posets with labels of n/4+o(n) bits per element. All these results are best possible, up to the lower order term.

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.

Abstract We resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $$ H_{1} $$ H 1 and $$H_2$$ H 2 for all but six pairs $$(H_1,H_2)$$ ( H 1 , H 2 ) . Schweitzer had previously shown that the number of open cases was finite, but without specifying the open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. Our work combines known results such as these with new results. By exploiting a relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for $$(H_1,H_2)$$ ( H 1 , H 2 ) -free graphs to five.

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

An 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 $(\frac{4}{3}+o(1))\log{n}$. All the labels of the input graph can be computed in polynomial time, while adjacency can be decided from the labels in constant time. 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 labeling scheme can be generalized to larger classes of topologically constrained graphs, for instance, to graphs embeddable in any fixed surface or to $k$-planar graphs for any fixed $k$, at the cost of larger second-order terms.

A k-coloring of a graph maps each vertex of the graph to a color in {1, 2, …, k}, such that no two adjacent vertices receive the same color. Given a k-coloring of a graph, a Kempe change produces a new k-coloring by swapping the colors in a bicolored connected component. We investigate the complexity of finding the smallest number of Kempe changes needed to transform a given k-coloring into another given k-coloring. We show that this problem admits a polynomial-time dynamic programming algorithm on path graphs, which turns out to be highly non-trivial. Furthermore, the problem is NP-hard even on star graphs and we show that on such graphs it admits a constant-factor approximation algorithm and is fixed-parameter tractable when parameterized by the number k of colors. The hardness result as well as the algorithmic results are based on the notion of a canonical transformation.

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this article we investigate this problem in graph classes defined by forbidding an induced subgraph. In particular, we provide output-polynomial time algorithms for K t -free graphs and for several related graph classes. This answers a question of Kanté et al. about enumeration in bipartite graphs.

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.

It is a long-standing open problem whether the minimal dominating sets of a graph can be enumerated in output-polynomial time. In this paper we prove that this is the case in triangle-free graphs. This answers a question of Kanté et al. Additionally, we show that deciding if a set of vertices of a bipartite graph can be completed into a minimal dominating set is a NP-complete problem.

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $$k\ge 0$$ , to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $$P_4$$ -free graphs. We show that it remains polynomial-time solvable for $$P_5$$ -free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most k for a given integer $$k\ge 0$$ . Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for H-free graphs with the complexity of 3-Colouring, Independent Odd Cycle Transversal and other related problems.

We propose a polynomial-time algorithm which takes as input a finite set of points of R^3 and computes, up to arbitrary precision, a maximum subset with diameter at most 1. More precisely, we give the first randomized EPTAS and deterministic PTAS for Maximum Clique in unit ball graphs. Our approximation algorithm also works on disk graphs with arbitrary radii, in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. Recently, it was shown that the disjoint union of two odd cycles is never the complement of a disk graph [Bonnet, Giannopoulos, Kim, Rzazewski, Sikora; SoCG '18]. This enabled the authors to derive a QPTAS and a subexponential algorithm for Max Clique on disk graphs. In this paper, we improve the approximability to a randomized EPTAS (and a deterministic PTAS). More precisely, we obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. We then address the question of computing Max Clique for disks in higher dimensions. We show that intersection graphs of unit balls, like disk graphs, do not admit the complement of two odd cycles as an induced subgraph. This, in combination with the first result, straightforwardly yields a randomized EPTAS for Max Clique on unit ball graphs. In stark contrast, we show that on ball graphs and unit 4-dimensional disk graphs, Max Clique is NP-hard and does not admit an approximation scheme even in subexponential-time, unless the Exponential Time Hypothesis fails.

Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the notion of distributed recoloring: The input graph represents a network of computers that needs to be recolored. Initially, each node is aware of its own input color and target color. The nodes can exchange messages with each other, and eventually each node has to stop and output its own recoloring schedule, indicating when and how the node changes its color. The recoloring schedules have to be globally consistent so that the graph remains properly colored at each point, and we require that adjacent nodes do not change their colors simultaneously. We are interested in the following questions: How many communication rounds are needed (in the deterministic LOCAL model of distributed computing) to find a recoloring schedule? What is the length of the recoloring schedule? And how does the picture change if we can use extra colors to make recoloring easier? The main contributions of this work are related to distributed recoloring with one extra color in the following graph classes: trees, 3-regular graphs, and toroidal grids.

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a forest. The set A in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

The NP-complete problem Feedback Vertex Set is to decide if it is possible, for a given integer k>=0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the 3-Colouring problem, or equivalently, to the problem of deciding if a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for H-free graphs. We prove that it is NP-complete if H contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for P_4-free graphs. We show that it remains in P for P_5-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks if a graph has an independent odd cycle transversal of size at most k for a given integer k>=0.

We continue research into a well-studied family of problems that ask if the vertices of a graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We let G be the class of k-degenerate graphs. The problem is known to be polynomial-time solvable if k=0 (bipartite graphs) and NP-complete if k=1 (near-bipartite graphs) even for graphs of diameter 4, as shown by Yang and Yuan, who also proved polynomial-time solvability for graphs of diameter 2. We show that recognizing near-bipartite graphs of diameter 3 is NP-complete resolving their open problem. To answer another open problem, we consider graphs of maximum degree D on n vertices. We show how to find A and B in O(n) time for k=1 and D=3, and in O(n^2) time for k >= 2 and D >= 4. These results also provide an algorithmic version of a result of Catlin [JCTB, 1979] and enable us to complete the complexity classification of another problem: finding a path in the vertex colouring reconfiguration graph between two given k-colourings of a graph of bounded maximum degree.

In the $$k$$ -Leaf Out-Branching and $$k$$ -Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is whether there exists an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems have been studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with $$O(k^2)$$ vertices are known on general graphs. In this work we show that $$k$$ -Leaf Out-Branching admits a kernel with O(k) vertices on $${{\mathcal {H}}}$$ -minor-free graphs, for any fixed family of graphs $${{\mathcal {H}}}$$ , whereas $$k$$ -Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.

Erdös and Hajnal conjectured that for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain an induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$ there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size at least $n^{c_k}$.

We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is $k$-edge-choosable if for any assignment of $k$ colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every graph is ($\Delta+1$)-edge-choosable. In 1990, Borodin solved the conjecture for planar graphs with maximum degree $\Delta\geq 9$ and asked whether the bound could be lowered to 8. We prove here that planar graphs with $\Delta\geq 8$ are ($\Delta+1$)-edge-choosable.