The {\it inversion} of a set $X$ of vertices in a digraph $D$ consists of reversing the direction of all arcs of $D\langle X\rangle$. We study $\sinv'_k(D)$ (resp. $\sinv_k(D)$) which is the minimum number of inversions needed to transform $D$ into a $k$-arc-strong (resp. $k$-strong) digraph and $\sinv'_k(n) = \max\{\sinv'_k(D) \mid D~\mbox{is a $2k$-edge-connected digraph of order $n$}\}$. We show : (i) $\frac{1}{2} \log (n - k+1) \leq \sinv'_k(n) \leq \log n + 4k -3$ for all $n \in \mathbb{Z}_{\geq 0}$; (ii) for any fixed positive integers $k$ and $t$, deciding whether a given oriented graph $\vec{G}$ satisfies $\sinv'_k(\vec{G}) \leq t$ (resp. $\sinv_k(\vec{G}) \leq t$) is NP-complete ; (iii) if $T$ is a tournament of order at least $2k+1$, then $\sinv'_k(T) \leq \sinv_k(T) \leq 2k$, and $\frac{1}{2}\log(2k+1) \leq \sinv'_k(T) \leq \sinv_k(T)$ for some $T$; (iv) if $T$ is a tournament of order at least $28k-5$ (resp. $14k-3$), then $\sinv_k(T) \leq 1$ (resp. $\sinv_k(T) \leq 6$); (v) for every $\epsilon>0$, there exists $C$ such that $\sinv_k(T) \leq C$ for every tournament $T$ on at least $2k+1 + \epsilon k$ vertices.

In this note, we introduce the partial order decomposition number of a digraph D, denoted pod(D), defined as the minimum integer k such that A(D)=A(P_1)∪⋯∪ A(P_k), where P_1,…,P_k are partial orders on V(D). We prove that (D)≤(D)^pod(D) for every digraph D. In particular, every class of digraphs with bounded pod is polynomially -bounded. We apply this to tournaments, showing that if 𝒞 is a class of tournaments with bounded dichromatic number, then the closure of 𝒞 under substitution is polynomially -bounded, thereby making progress on a question of Aubian, Charbit, Lopes, and the first author. As further applications of pod, we prove that poset tournaments of bounded dimension are -bounded, derive polynomial lower bounds on the directed clique number of an explicit family of tournaments, thereby answering a conjecture of Gutowski and Rams, and show that tournaments with bounded pod have bounded domination number.

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: ∙ All-Pairs Shortest Paths, ∙ the multiplication of the adjacency matrix M of the input graph with any n × 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, Thomassé; 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.

For every ε > 0, Max Independent Set admits a polynomial-time n^ε-approximation algorithm on n-vertex graphs of effectively bounded twin-width [Bergé et al., STACS '23]. The approximation factor actually obtained is more precisely n^O(1/ loglog 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 γ> 0 such that a polynomial-time n^γ/ (loglog 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/ loglog 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 Toruńczyk, 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 cml^↓, 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 Maximum Induced d-Regular Subgraph and Induced Disjoint Paths in graphs given with a contraction sequence witnessing low cml^↓, 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 cml^↓: 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 cml^↓ 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^↓ (including reduced bandwidth), the family of classes of bounded p^↓ 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.

Dallard, Milanič, and Štorgel [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 ⊎ C_4 (the disjoint union of t triangles and a 4-vertex cycle). The former generalizes the algorithm readily obtained from Alekseev’s structural result on graphs excluding tK_2 as an induced subgraph [Alekseev, DAM ’07], while the latter extends Bonamy et al.’s result.

We show that it is NP-hard to distinguish graphs of linear mim-width at most 1211 from graphs of sim-width at least 1216. This implies that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear counterparts are all paraNP-complete, i.e., NP-complete to compute even when upper bounded by a constant.

A graph class admits an implicit representation if, for every positive integer n, its n-vertex graphs have a O(log n)-bit (adjacency) labeling scheme, i.e., their vertices can be labeled by binary strings of length O(log n) such that the presence of an edge between any pair of vertices can be deduced solely from their labels. The famous Implicit Graph Conjecture posited that every hereditary (i.e., closed under taking induced subgraphs) factorial (i.e., containing 2(O( n log n)) n-vertex graphs) class admits an implicit representation. The conjecture was recently refuted [Hatami and Hatami, FOCS '22], and does not even hold among monotone (i.e., closed under taking subgraphs) factorial classes [Bonnet et al., ICALP '24]. However, monotone small (i.e., containing at most n!c(n) many n-vertex graphs for some constant c) classes do admit implicit representations. This motivates the Small Implicit Graph Conjecture: Every hereditary small class admits an O(log n)-bit labeling scheme. We provide evidence supporting the Small Implicit Graph Conjecture. First, we show that every small weakly sparse (i.e., excluding some fixed bipartite complete graph as a subgraph) class has an implicit representation. This is a consequence of the following fact of independent interest proved in the paper: Every weakly sparse small class has bounded expansion (hence, in particular, bounded degeneracy). Second, we show that every hereditary small class admits an O(log(3) n)-bit labeling scheme, which provides a substantial improvement of the best-known polynomial upper bound of n(1-epsilon) on the size of adjacency labeling schemes for such classes. This is a consequence of another fact of independent interest proved in the paper: Every small class has neighborhood complexity O( n log n).

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 $n^{O(t^2 \log n)+t^{O(1)}}$ 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.

A class of graphs admits an adjacency labeling scheme of size $b(n)$, if the vertices in each of its $n$-vertex graphs can be assigned binary strings (called labels) of length $b(n)$ so that the adjacency of two vertices can be determined solely from their labels. We give tight bounds on the size of adjacency labels for every family of monotone (i.e., subgraph-closed) classes with a well-behaved growth function between $2^{O(n \log n)}$ and $2^{O(n^{2-\delta})}$ for any $\delta > 0$. Specifically, we show that for any function $f: \mathbb N \to \mathbb R$ satisfying $\log n \leqslant f(n) \leqslant n^{1-\delta}$ for any fixed $\delta > 0$, and some~sub-multiplicativity condition, there are monotone graph classes with growth $2^{O(nf(n))}$ that do not admit adjacency labels of size at most $f(n) \log n$. On the other hand, any such class does admit adjacency labels of size $O(f(n)\log n)$. Surprisingly this tight bound is a $\Theta(\log n)$ factor away from the information-theoretic bound of $\Omega(f(n))$. The special case when $f = \log$ implies that the recently-refuted Implicit Graph Conjecture [Hatami and Hatami, FOCS 2022] also fails within monotone classes. We further show that the Implicit Graph Conjecture holds for all monotone \emph{small} classes. In other words, any monotone class with growth rate at most $n!\,c^n$ for some constant $c>0$, admits adjacency labels of information-theoretic order optimal size. In fact, we show a more general result that is of independent interest: any monotone small class of graphs has bounded degeneracy.We conjecture that the Implicit Graph Conjecture holds for all hereditary small classes.

We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most d if it admits an elimination order of its vertices where a vertex u can be removed whenever it has a d-twin, i.e., another vertex v such that at most d vertices outside {u,v} are neighbors of exactly one of u, v. The family of graph classes of bounded sd-degeneracy is a superset of that of graph classes of bounded degeneracy or of bounded flip-width, and more generally, of bounded symmetric difference. Unlike most graph parameters, sd-degeneracy is not hereditary: it may be strictly smaller on a graph than on some of its induced subgraphs. In particular, every n-vertex graph is an induced subgraph of some O(n^2)-vertex graph of sd-degeneracy 1. In spite of this and the breadth of classes of bounded sd-degeneracy, we devise Õ(√(n))-bit adjacency labeling schemes for them, which are optimal up to the hidden polylogarithmic factor. This is attained on some even more general classes, consisting of graphs G whose vertices bijectively map to the leaves of a tree T, where transversal edges and anti-edges added to T define the edge set of G. We call such graph representations signed tree models as they extend the so-called tree models (or twin-decompositions) developed in the context of twin-width, by adding transversal anti-edges. While computing the degeneracy of an input graph can be done in linear time, we show that deciding whether its symmetric difference is at most 8 is co-NP-complete, and whether its sd-degeneracy is at most 1 is NP-complete.

We show that for any natural number s, there is a constant γ and a subgraph-closed class having, for any natural n, at most γ^n graphs on n vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most s log n. In other words, for every s, there is a small (even tiny) monotone class without universal graphs of size n^s. Prior to this result, it was not excluded that every small class has an almost linear universal graph, or equivalently a labeling scheme with labels of size (1+o(1))log n. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, ESA '07; Dujmović et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small monotone classes have unbounded twin-width, thus simultaneously disprove the already-refuted Small conjecture; but this time with a self-contained proof, not relying on elaborate group-theoretic constructions.

We construct a hereditary class of triangle-free graphs with unbounded chromatic number, in which every non-trivial graph either contains a pair of non-adjacent twins or has an edgeless vertex cutset of size at most two. This answers in the negative a question of Chudnovsky, Penev, Scott, and Trotignon. The class is the hereditary closure of a family of (triangle-free) twincut graphs $G_1, G_2, \ldots$ such that $G_k$ has chromatic number $k$. We also show that every twincut graph is edge-critical.

A perfect matching cut is a perfect matching that is also a cutset, or equivalently a perfect matching containing an even number of edges on every cycle. The corresponding algorithmic problem, Perfect Matching Cut, is known to be NP-complete in subcubic bipartite graphs [Le Telle, TCS '22] but its complexity was open in planar graphs and in cubic graphs. We settle both questions at once by showing that Perfect Matching Cut is NP-complete in 3-connected cubic bipartite planar graphs or Barnette graphs. Prior to our work, among problems whose input is solely an undirected graph, only Distance-2 4-Coloring was known NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette's conjecture were refuted.

We introduce a new parameter, called stretch-width, that we show sits strictly between clique-width and twin-width. Unlike the reduced parameters [BKW '22], planar graphs and polynomial subdivisions do not have bounded stretch-width. This leaves open the possibility of efficient algorithms for a broad fragment of problems within Monadic Second-Order (MSO) logic on graphs of bounded stretch-width. In this direction, we prove that graphs of bounded maximum degree and bounded stretch-width have at most logarithmic treewidth. As a consequence, in classes of bounded stretch-width, Maximum Independent Set can be solved in subexponential time $2^{O(n^{4/5} \log n)}$ on $n$-vertex graphs, and, if further the maximum degree is bounded, Existential Counting Modal Logic [Pilipczuk '11] can be model-checked in polynomial time. We also give a polynomial-time $O(\text{OPT}^2)$-approximation for the stretch-width of symmetric $0,1$-matrices or ordered graphs. Somewhat unexpectedly, we prove that exponential subdivisions of bounded-degree graphs have bounded stretch-width. This allows to complement the logarithmic upper bound of treewidth with a matching lower bound. We leave as open the existence of an efficient approximation algorithm for the stretch-width of unordered graphs, if the exponential subdivisions of all graphs have bounded stretch-width, and if graphs of bounded stretch-width have logarithmic clique-width (or rank-width).

We show that for any natural number $s$, there is a constant $\gamma$ and a subgraph-closed class having, for any natural $n$, at most $\gamma^n$ graphs on $n$ vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most $s \log n$. In other words, for every $s$, there is a small (even tiny) monotone class without universal graphs of size $n^s$. Prior to this result, it was not excluded that every small class has an almost linear universal graph, or equivalently a labeling scheme with labels of size $(1+o(1))\log n$. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, ESA '07; Dujmovi\'{c} et al., JACM '21; Bonamy et al., SIDMA '22; Bonnet et al., Comb. Theory '22]. Furthermore, our small monotone classes have unbounded twin-width, thus simultaneously disprove the already-refuted Small conjecture; but this time with a self-contained proof, not relying on elaborate group-theoretic constructions.