arXiv subjects
Publications and source records attributed to Édouard Bonnet.
We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.
The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomass\'e; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.
We prove that every finite point set of size at least $10^{11055931}$ has four collinear points or six points that pairwise see each other. This resolves the first open case of the big-line-big-clique conjecture of K\'ara, P\'or, and Wood.
For every $\varepsilon > 0$, it is NP-hard to $n^{1-\varepsilon}$-approximate Max Independent Set in $n$-vertex graphs [Hastad '96, Zuckerman '07]. In triangle-free graphs, a simple argument gives a polynomial-time $n^{1/2}$-approximation algorithm, whereas, for every $\varepsilon > 0$, an $n^{1/4-\varepsilon}$-approximation algorithm would imply that NP $\subseteq$ BPP [Bonnet, Thomass\'e, Tran, Watrigant; ESA '20]. In this note, we close this gap by proving the corresponding hardness against $n^{1/2-\varepsilon}$-approximation algorithms. The reduction is very simple and uses the Moser-Tardos resampling algorithm to make the constructed graphs triangle-free. The soundness uses a result of Haeupler, Saha, and Srinivasan building on the proof of Moser and Tardos, to upper-bound the probability that a fixed relatively large subset is an independent set after the Moser-Tardos algorithm terminates. We generalize this scheme and show that, for any nonempty finite family $\mathcal F$ of graphs, each containing at least one cycle, for any $\varepsilon > 0$, an $n^{\mu(\mathcal F)-\varepsilon}$-approximation algorithm for Max Independent Set in graphs excluding every member of $\mathcal F$ as a subgraph implies that NP $\subseteq$ BPP, where $\mu(\mathcal F) := 1 - \max\limits_{H \in \mathcal F}~\min\limits_{U \subseteq V(H), H[U] \text{contains a cycle}} (|U|-2)/(|E(H[U])|-1)$.
Given a connected graph $G$ and a source $s \in V(G)$, what is the smallest number of rounds necessary for all vertices of $G$ to receive a message initially only held by $s$, where at each round every informed vertex passes the message to one of its neighbors? This problem is called Telephone Broadcast and is suprisingly hard: it remains NP-hard on cycles intersecting at a single shared vertex, in particular, graphs of pathwidth 2 with a linear feedback vertex set of size 1, as well as on graphs with treedepth at most 6 [Egami et al.; MFCS '25]. Vertex cover number, vertex integrity, and distance to clique are among the few parameters for which Telephone Broadcast is fixed-parameter tractable. There is a $2^{\mathcal{O}(\mathrm{vc}^3)} n^{\mathcal{O}(1)}$-time algorithm parameterized by vertex cover number $\mathrm{vc}$ [Fomin, Fraigniaud, Golovach; TCS '24], a double-exponential algorithm parameterized by vertex integrity $\mathrm{vi}$, and a $2^{\mathcal{O}(k^2)} n^{\mathcal{O}(1)}$-time algorithm parameterized by distance to clique $k$ [Egami et al.; MFCS '25]. In this paper, we give improved parameterized algorithms for Telephone Broadcast with running times $2^{\mathcal{O}(\mathrm{vc} \log \mathrm{vc})} n^{\mathcal{O}(1)}$, $2^{\mathcal{O}(\mathrm{vi}^2 \log \mathrm{vi})} n^{\mathcal{O}(1)}$, and $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$. The main ingredient that makes our algorithms faster is a Turing reduction to edge-weighted $b$-Matching.
For every $\varepsilon > 0$, Max Independent Set admits a polynomial-time $n^\varepsilon$-approximation algorithm on $n$-vertex graphs of effectively bounded twin-width [Berg\'e et al., STACS '23]. The approximation factor actually obtained is more precisely $n^{O(1/ \log \log n)}$. Prior to the current paper, no approximation hardness was known for this problem, and the existence of a polynomial-time approximation scheme (PTAS) was repeatedly raised as an open question. We answer this question in a strong sense: We show that there is a constant $\gamma > 0$ such that a polynomial-time $n^{\gamma/ (\log \log n)^2}$-approximation algorithm for Max Independent Set on graphs of twin-width at most 4 would refute the Exponential-Time Hypothesis (ETH). This lower bound further holds if a 4-sequence is provided as part of the input. We show the same hardness of approximation for Min Coloring, which also has a nearly matching $n^{O(1/ \log \log n)}$-approximation algorithm on graphs of effectively bounded twin-width. We also clarify the parameterized complexity of $k$-Independent Set on graphs of bounded radius-$r$ merge-width when the range of $r$ is limited. There is a fixed-parameter tractable algorithm for $k$-Independent Set on graphs given with radius-$2^{O(k^2)}$ merge sequences of bounded width [Dreier and Toru\'nczyk, STOC '25]. We complement this result by showing that $k$-Independent Set is W[1]-hard on graphs given with radius-$o(k)$ merge sequences of bounded width. We further show that this result also holds for $k$-Dominating Set.
Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by $\operatorname{cml}^\downarrow$, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as \textsc{Maximum Induced $d$-Regular Subgraph} and \textsc{Induced Disjoint Paths} in graphs given with a contraction sequence witnessing low $\operatorname{cml}^\downarrow$, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs. We get the following collapses in sparse classes of bounded $\operatorname{cml}^\downarrow$: bounded maximum degree implies bounded treewidth, whereas $K_{t,t}$-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded $\operatorname{cml}^\downarrow$ admit balanced separators dominated by a bounded number of vertices. We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters $p^\downarrow$ (including reduced bandwidth), the family of classes of bounded $p^\downarrow$ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids.
Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.
We present a quasipolynomial-time approximation scheme (QPTAS) for the Maximum Independent Set (\textsc{MWIS}) in graphs with a bounded number of pairwise vertex-disjoint and non-adjacent long induced cycles. More formally, for every fixed $s$ and $t$, we show a QPTAS for \textsc{MWIS} in graphs that exclude $sC_t$ as an induced minor. Combining this with known results, we obtain a QPTAS for the problem of finding a largest induced subgraph of bounded treewidth with given hereditary property definable in Counting Monadic Second Order Logic, in the same classes of graphs. This is a step towards a conjecture of Gartland and Lokshtanov which asserts that for any planar graph $H$, graphs that exclude $H$ as an induced minor admit a polynomial-time algorithm for the latter problem. This conjecture is notoriously open and even its weaker variants are confirmed only for very restricted graphs $H$.
A signed tree model of a graph $G$ is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of $G$, together with 2-colored edges $xy$, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at $x$ and at $y$. We design an algorithm that, given such a representation of an $n$-vertex graph $G$ with $p$ transversal pairs and a source $v \in V(G)$, computes a shortest-path tree rooted at $v$ in $G$ in time $O(p \log n)$. A wide variety of graph classes are such that for all $n$, their $n$-vertex graphs admit signed tree models with $O(n)$ transversal pairs: for instance, those of bounded symmetric difference, more generally of bounded sd-degeneracy, as well as interval graphs. As applications of our Single-Source Shortest Path algorithm and new techniques, we - improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier and Toru\'nczyk, STOC '25] to quadratic; - give an $O(n^2 \log n)$-time algorithm for All-Pairs Shortest Path (APSP) on graphs given with a witness of low merge-width, generalizing a result known on twin-width [Twin-Width III, SICOMP '24]; - extend and simplify an $O(n^2 \log n)$-time algorithm for multiplying two $n \times n$ matrices $A, B$ of bounded twin-width in [Twin-Width V, STACS '23]: now $A$ solely has to be an adjacency matrix of a graph of bounded twin-width and $B$ can be arbitrary; - give an $O(n^2 \log^2 n)$-time algorithm for APSP on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24]; - give an $O(n^{7/3} \log^2 n)$-time algorithm for APSP on graphs of symmetric difference $O(n^{1/3})$.
As the class $\mathcal T_4$ of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an $n$-vertex graph at least $2 \log n$ times, most NP-hard graph problems, like Max Independent Set, Dominating Set, Hamiltonian Cycle, remain so on $\mathcal T_4$. However, Min Coloring and k-Coloring are easy on both families because they are 2-colorable and 3-colorable, respectively. We show that Min Coloring is NP-hard on the class $\mathcal T_3$ of graphs of twin-width at most 3. This is the first hardness result on $\mathcal T_3$ for a problem that is easy on cographs (twin-width 0), on trees (whose twin-width is at most 2), and on unit circular-arc graphs (whose twin-width is at most 3). We also show that for every $k \geqslant 3$, k-Coloring is NP-hard on $\mathcal T_4$. We finally make two observations: (1) there are currently very few problems known to be in P on $\mathcal T_d$ (graphs of twin-width at most $d$) and NP-hard on $\mathcal T_{d+1}$ for some nonnegative integer $d$, and (2) unlike $\mathcal T_4$, which contains every graph as an induced minor, the class $\mathcal T_3$ excludes a fixed planar graph as an induced minor; thus it may be viewed as a special case (or potential counterexample) for conjectures about classes excluding a (planar) induced minor. These observations are accompanied by several open questions.
In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been devoted to understanding the unavoidable induced substructures of graphs with large pathwidth or large treewidth. In this paper, we give an induced counterpart of the Forest Minor theorem: for any $t \geqslant 2$, the $K_{t,t}$-subgraph-free $H$-induced-minor-free graphs have bounded pathwidth if and only if $H$ belongs to a class $\mathcal F$ of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs $H$ for which every weakly sparse $H$-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.
The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.
Treedepth is a central parameter to algorithmic graph theory. The current state-of-the-art in computing and approximating treedepth consists of a $2^{O(k^2)} n$-time exact algorithm and a polynomial-time $O(\text{OPT} \log^{3/2} \text{OPT})$-approximation algorithm, where the former algorithm returns an elimination forest of height $k$ (witnessing that treedepth is at most $k$) for the $n$-vertex input graph $G$, or correctly reports that $G$ has treedepth larger than $k$, and $\text{OPT}$ is the actual value of the treedepth. On the complexity side, exactly computing treedepth is NP-complete, but the known reductions do not rule out a polynomial-time approximation scheme (PTAS), and under the Exponential Time Hypothesis (ETH) only exclude a running time of $2^{o(\sqrt n)}$ for exact algorithms. We show that 1.0003-approximating treedepth is NP-hard, and that exactly computing the treedepth of an $n$-vertex graph requires time $2^{\Omega(n)}$, unless the ETH fails. We further derive that there exist absolute constants $\delta, c > 0$ such that any $(1+\delta)$-approximation algorithm requires time $2^{\Omega(n / \log^c n)}$. We do so via a simple direct reduction from Satisfiability to Treedepth, inspired by a reduction recently designed for Treewidth [STOC '25].
A graph class $\mathcal C$ is monadically dependent if one cannot interpret all graphs in colored graphs from $\mathcal C$ using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every $r\in \mathbb{N}$, $\varepsilon>0$, and every graph $G\in \mathcal{C}$ equipped with a weight function on vertices, one can apply a bounded (in terms of $\mathcal{C},r,\varepsilon$) number of flips (complementations of the adjacency relation on a subset of vertices) to $G$ so that in the resulting graph, every radius-$r$ ball contains at most an $\varepsilon$-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.
We show that for any positive integers $g$ and $t$, there is a $K_{6}^{(1)}$-induced-minor-free graph of girth at least $g$ that is not a region intersection graph over the class of $K_t$-minor-free graphs. This answers in a strong form the recently raised question of whether for every graph $H$ there is a graph $H'$ such that $H$-induced-minor-free graphs are region intersection graphs over $H'$-minor-free graphs.
We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth. This refutes several conjectures and critically thwarts the quest for the unavoidable induced subgraphs in classes of unbounded treewidth, a wished-for counterpart of the Grid Minor theorem. We actually show a stronger result: For every positive integer $t$, there is a hereditary graph class $\mathcal C_t$ of unbounded treewidth such that for any graph $H$ of treewidth at most $t$, the $H$-free graphs of $\mathcal C_t$ have bounded treewidth. Our construction is a variant of so-called layered wheels. We also introduce a framework of abstract layered wheels, based on their most salient properties. In particular, we streamline and extend key lemmas previously shown on individual layered wheels. We believe that this should greatly help develop this topic, which appears to be a very strong yet underexploited source of counterexamples.
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 $2^{\Omega(\sqrt n)}$. 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.