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 Independent Set, Maximum Clique, 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. In contrast, there are graphs G of arbitrarily large girth and treewidth Θ(|V(G)|^{1/2}) such that cml^↓(G) ⩽ 3. 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; HJ, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids.

Abstract. Consider a graph [Formula: see text] with a long path [Formula: see text]. When is it the case that [Formula: see text] also contains a long induced path? This question has been investigated in general as well as within a number of different graph classes since the 1980s. We have recently observed in a companion paper [ Long induced paths in sparse graphs and graphs with forbidden patterns, preprint, arXiv:2411.08685, 2024] that most existing results can be recovered in a simple way by considering forbidden ordered patterns of edges along the path [Formula: see text]. In particular, we proved that if we forbid some fixed ordered matching along a path of order [Formula: see text] in a graph [Formula: see text], then [Formula: see text] must contain an induced path of order [Formula: see text]. Moreover, we completely characterized the forbidden ordered patterns forcing the existence of an induced path of polynomial size. The purpose of the present paper is to completely characterize the ordered patterns [Formula: see text] such that forbidding [Formula: see text] along a path [Formula: see text] of order [Formula: see text] implies the existence of an induced path of order [Formula: see text]. These patterns are star forests with some specific ordering, which we call constellations. As a direct consequence of our result, we show that if a graph [Formula: see text] has a path of length [Formula: see text] and does not contain [Formula: see text] as a topological minor, then [Formula: see text] contains an induced path of order [Formula: see text]. The previously best known bound was [Formula: see text] for some unspecified function [Formula: see text] depending on the Topological Minor Structure Theorem of Grohe and Marx (2015).

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 key intermediate problem that we introduce and show NP-complete, Linear Degree Balancing, inputs an edge-weighted graph G and an integer τ, and asks whether V(G) can be linearly ordered such that every vertex of G has weighted backward and forward degrees at most τ.

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 $n^{1-\varepsilon}$ 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 $O(n \log n)$.

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 $2^{O(n^{2-δ})}$ for any $δ> 0$. Specifically, we show that for any function $f: \mathbb N \to \mathbb R$ satisfying $\log n \leqslant f(n) \leqslant n^{1-δ}$ for any fixed $δ> 0$, 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 $Θ(\log n)$ factor away from the information-theoretic bound of $Ω(f(n))$. 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(n²)-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 a graph takes linear time, we show that determining its symmetric difference is para-co-NP-complete. This may seem surprising as symmetric difference can serve as a short-sighted first approximation of twin-width, whose computation is para-NP-complete. Indeed, we show that deciding if the symmetric difference of an input graph is at most 8 is co-NP-complete. We also show that deciding if the sd-degeneracy is at most 6 is NP-complete, contrasting with the symmetric difference.

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 γn 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 ns. 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.

Abstract. We show that for any natural number [Formula: see text], there is a constant [Formula: see text] and a subgraph-closed class having, for any natural [Formula: see text], at most [Formula: see text] graphs on [Formula: see text] vertices up to isomorphism, but no adjacency labeling scheme with labels of size at most [Formula: see text]. In other words, for every [Formula: see text], there is a small—even tiny—monotone class without universal graphs of size [Formula: see text]. 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 [Formula: see text]. The existence of such a labeling scheme, a scaled-down version of the recently disproved Implicit Graph Conjecture, was repeatedly raised [Gavoille and Labourel, Proceedings of the 15 th Annual European Symposium on Algorithms, Lecture Notes in Comput. Sci. 4698, Springer, 2007, pp. 582–593; Dujmović et al., J. ACM, 68 (2021), pp. 1–33; Bonamy, Gavoille, and Pilipczuk, SIAM J. Discrete Math., 36 (2022), pp. 2082–2099; Bonnet et al., Comb. Theory, 2 (2022)]. Furthermore, our small monotone classes have unbounded twin-width and thus simultaneously disprove the already-refuted Small conjecture, but this time with a self-contained proof, not relying on elaborate group-theoretic constructions. As our main ingredient, we show that with high probability an Erdős–Rényi random graph [Formula: see text] with [Formula: see text] has, for every [Formula: see text], at most [Formula: see text] subgraphs on [Formula: see text] vertices, up to isomorphism. As a barrier to our general method of producing even more complex tiny classes, we show that when [Formula: see text], the latter no longer holds. More concretely, we provide an explicit lower bound on the number of unlabeled [Formula: see text]-vertex induced subgraphs of [Formula: see text] when [Formula: see text]. We thereby obtain a threshold for the property of having exponentially many unlabeled induced subgraphs: if [Formula: see text] with [Formula: see text], then with high probability even the number of all unlabeled (not necessarily induced) subgraphs is [Formula: see text], whereas if [Formula: see text] for sufficiently large [Formula: see text], then with high probability the number of unlabeled induced subgraphs is [Formula: see text]. This result supplements the study of counting unlabeled induced subgraphs that was initiated by Erdős and Rényi with a question on the number of unlabeled induced subgraphs of Ramsey graphs, eventually answered by Shelah.

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 cubic graphs. We settle both questions simultaneously 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 to be NP-complete in Barnette graphs. Notably, Hamiltonian Cycle would only join this private club if Barnette's conjecture were refuted. Funding This work was supported by the ANR projects TWIN-WIDTH (ANR-21-CE48-0014) and Digraphs (ANR-19-CE48-0013). Acknowledgements We are much indebted to Carl Feghali for introducing us to the topic of (perfect) matching cuts, and for presenting open problems to us that led to the current paper. We also wish to thank him and Kristóf Huszár for helpful discussions at an early stage of the project.

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^{Õ(n^{8/9})} 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(OPT²)-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).