Given an integer n, let G(n) be the number of integer sequences n − 1 ≥ d₁ ≥ d₂ ≥ · · · ≥ d_n ≥ 0 that are the degree sequence of some graph. We show that G(n) = (c + o(1))4ⁿ/n³/⁴ for some constant c > 0, improving both the previously best upper and lower bounds by a factor of n¹/⁴ (up to polylog-factors). Additionally, we answer a question of Royle, extend the values of n for which G(n) is known exactly from n ≤ 290 to n ≤ 1651 and determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative.

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for (Formula presented.) in every (Formula presented.) -connected graph any two longest cycles intersect in at least (Formula presented.) vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph, which can be used to improve the best known bounds toward all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on (Formula presented.) vertices contains a cycle (and hence path) of length at least (Formula presented.), improving on (Formula presented.) from DeVos [arXiv:2302:04255, 2023]. Second, we show that in every (Formula presented.) -connected graph with (Formula presented.), any two longest cycles meet in at least (Formula presented.) vertices, improving on (Formula presented.) from Chen, Faudree, and Gould [J. Combin. Theory, Ser. B, 72 (1998) no. 1, 143–149]. Our proof combines combinatorial arguments, computer search, and linear programming.

Given access to the vertex set (Formula presented.) of a connected graph (Formula presented.) and an oracle that given two vertices (Formula presented.), returns the shortest path distance between (Formula presented.) and (Formula presented.), how many queries are needed to reconstruct (Formula presented.) ? Firstly, we show that randomized algorithms need to use at least (Formula presented.) queries in expectation to reconstruct (Formula presented.) -vertex trees of maximum degree (Formula presented.). The best previous lower bound (for graphs of bounded maximum degree) was an information-theoretic lower bound of (Formula presented.). Our randomized lower bound is also the first to break through the information-theoretic barrier for related query models, including distance queries for phylogenetic trees, membership queries for learning partitions, and path queries in directed trees. Secondly, we provide a simple deterministic algorithm to reconstruct trees using (Formula presented.) distance queries. This proves that our lower bound is optimal up to a multiplicative constant. We extend our algorithm to reconstruct graphs without induced cycles of length at least (Formula presented.) using (Formula presented.) queries. Our lower bound is therefore tight for a wide range of tree-like graphs, such as chordal graphs, permutation graphs, and AT-free graphs. The previously best randomized algorithm for chordal graphs used (Formula presented.) queries in expectation, so we improve by a (Formula presented.) -factor for this graph class.

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in O(n⁴) time, where n denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.

A homomorphism from a graph G to a graph H is an edge-preserving mapping from V (G) to V (H). In the graph homomorphism problem, denoted by Hom(H), the graph H is fixed and we need to determine if there exists a homomorphism from an instance graph G to H. We study the complexity of the problem parameterized by the cutwidth of G, i.e., we assume that G is given along with a linear ordering v1, . . ., vn of V (G) such that, for each i ∈ {1, . . ., n − 1}, the number of edges with one endpoint in {v1, . . ., vi} and the other in {vi+1, . . ., vn} is at most k. We aim, for each H, for algorithms for Hom(H) running in time c^k_Hn^O(1) and matching lower bounds that exclude c^k_H^·o⁽¹⁾n^O(1) or c^k_H^(1−Ω(1))n^O(1) time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call mimsup(H). Our main contribution is strong evidence of a close connection between cH and mimsup(H): an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most mimsup(H)^k, lower bounds that show that for almost all graphs H indeed we have cH ≥ mimsup(H), assuming the (Strong) Exponential-Time Hypothesis, and an algorithm with running time exp(O(mimsup(H) · k log k))n^O(1). In the last result we do not need to assume that H is a fixed graph. Thus, as a consequence, we obtain that the problem of deciding whether G admits a homomorphism to H is fixed-parameter tractable, when parameterized by cutwidth of G and mimsup(H). The parameter mimsup(H) can be thought of as the p-th root of the maximum induced matching number in the graph obtained by multiplying p copies of H via a certain graph product, where p tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of H. Such parameters play a central role in, among others, algebraic complexity theory and additive combinatorics. Our results tightly link the parameterized complexity of a problem to such an asymptotic matrix parameter for the first time.

Given a finite poset P, we say that a family F of subsets of [n] is P-saturated if F does not contain an induced copy of P, but adding any other set to F creates an induced copy of P. The induced saturation number of P, denoted by sat^∗(n,P), is the size of the smallest P-saturated family with ground set [n]. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that sat^∗(n,P)=O(n^c), where c≤|P|²/4+1 is a constant depending on P only. We obtain this result by bounding the VC-dimension of our family.

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices u and v, the oracle returns the shortest path distance between u and v in the graph. The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an n-vertex connected graph G parameterized by maximum degree ∆ and treelength k in O_k∆(nlog² n) queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.

Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f(k)n^O(1) and space f(k)log⁡(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as LIST COLORING and PRECOLORING EXTENSION with pathwidth as parameter, SCHEDULING OF JOBS WITH PRECEDENCE CONSTRAINTS, with both number of machines and partial order width as parameter, BANDWIDTH and variants of WEIGHTED CNF-SATISFIABILITY. In particular, this implies that all these problems are W[t]-hard for all t.

An independent transversal of a graph (Formula presented.) with a vertex partition (Formula presented.) is an independent set of (Formula presented.) intersecting each block of (Formula presented.) in a single vertex. Wanless and Wood proved that if each block of (Formula presented.) has size at least (Formula presented.) and the average degree of vertices in each block is at most (Formula presented.), then an independent transversal of (Formula presented.) exists. We present a construction showing that this result is optimal: for any (Formula presented.) and sufficiently large (Formula presented.), there is a forest with a vertex partition into parts of size at least (Formula presented.) such that the average degree of vertices in each block is at most (Formula presented.), and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld–Wanless–Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.

Two permutations σ and π are ℓ-similar if they can be decomposed into subpermutations σ(1), . . . ,σ(ℓ) and π(1), . . . ,π(ℓ) such that σ(i) is order-isomorphic to π(i) for all i ∈[ℓ]. Recently, Dudek, Grytczuk, and Ruciński Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum ℓ for which two permutations chosen independently and uniformly at random are ℓ-similar. We show that two such permutations are O(n1/3 log11/6(n))-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.

A classic model to study strategic decision making in multi-agent systems is the normal-form game. This model can be generalised to allow for an infinite number of pure strategies leading to continuous games. Multi-objective normal-form games are another generalisation that model settings where players receive separate payoffs in more than one objective. We bridge the gap between the two models by providing a theoretical guarantee that a game from one setting can always be transformed to a game in the other. We extend the theoretical results to include guaranteed equivalence of Nash equilibria. The mapping makes it possible to apply algorithms from one field to the other. We demonstrate this by introducing a fictitious play algorithm for multi-objective games and subsequently applying it to two well-known continuous games. We believe the equivalence relation will lend itself to new insights by translating the theoretical guarantees from one formalism to another. Moreover, it may lead to new computational approaches for continuous games when a problem is more naturally solved in the succinct format of multi-objective games.

We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of (t√log t) This is the first algorithm to achieve an f(t)-approximation for some function f. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant c such that every graph with pathwidth ω(kc) has treewidth at least k or contains a subdivision of a complete binary tree of height k. Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t′ in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d + 1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games. We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ⊆ S[t] ⊆ A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems.

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.

We show that List Colouring can be solved on n-vertex trees by a deterministic Turing machine using O(log n) bits on the worktape. Given an n-vertex graph G = (V,E) and a list L(v) ⊆ {1, . . . , n} of available colours for each v ∈ V , a list colouring for G is a proper colouring c such that c(v) ∈ L(v) for all v.

The deck of a graph G is the multiset of cards {G−v:v∈V(G)}. Myrvold (1992) showed that the degree sequence of a graph on n≥7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can be reconstructed from any deck missing O(n/d³) cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k⁷) for G or reports that G has tree-partition width more than k, in time k^O(1)n². We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.

In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k) log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on “tree-structured graphs” are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a “natural home” for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k) log n space. Moreover, we introduce “tree-shaped” variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.