Papers

A de Bruijn sequence packs all $n$-bit binary words into a cycle of length $2^n$. A de Bruijn torus is the two-dimensional analogue in which each word appears exactly once in a rectangular window. Here we consider the natural analogue for permutations using their shorthand representation (i.e., each permutation's final redundant value is omitted from the window). We show that these tori exist when $n = 2m + 1$ is odd and the torus and windows have two rows (i.e., the torus is a "tworus"). These twori can be constructed with a high degree of symmetry. More specifically, there are twori that can be partitioned into $2^{m-1}$ matching blocks where each block contains the same sequence of unordered columns. Furthermore, given one such block we can generate each successive column of a tworus in amortized $\mathcal{O}(1)$-time. We also prove non-existence results for certain sizes of tori and provide algorithms for constructing multiversal cycles (perfect necklaces) of unlabeled binary words.

We prove that the Turán density of the tetrahedron $K_4^{(3)}$ satisfies $\pi(K_4^{(3)}) \le 14993367693127837/26880000000000000 < 0.557789$, improving Baber's upper bound of $0.5615$ and closing about $62\%$ of the gap to the conjectured value $5/9$. The proof uses an exact seven-vertex flag-algebra certificate incorporating degree-stationarity from Razborov's differential method. To find the certificate, we combine the established techniques of cutting planes and column generation to optimize jointly over flag families whose types have at most five vertices. We give a complete formal proof of this Turán density bound in Lean 4.

We give snakes that are longer than the previous best known in dimensions 13 through 20 and improve the general lower bound for every dimension $d \geq 21$. Our explicit snakes reach 371,711 edges in dimension 20. Twenty compatible paths in that cube allow generalisation to give snakes of length at least $(17/48)2^d$ for every $d \geq 21$. Their controlled overlaps allow copies to be joined across the layers of a larger cube without creating shortcuts. Four additional paths give the same bound for coils. We explain the construction, prove the joining rule, and then count its length. The finite paths and their required intersections are independently verifiable.

A graph is 1-planar if it has a drawing in which every edge is crossed at most once. We show that the smallest cubic non-1-planar graphs have $30$ vertices. Two such graphs are the Tutte-Coxeter graph of girth eight and a graph of girth seven that we call the Byte graph. Every subcubic graph with fewer than $30$ vertices is 1-planar. Our proof is computer-assisted, but directly testing all relevant graphs is impractical. To establish non-1-planarity of the two graphs, we extend a SAT-based solver with a custom clause propagator based on separating cycles and a case split based on graph automorphisms, allowing independent cases to be solved in parallel. To show that all smaller subcubic graphs are 1-planar, we introduce the concept of $k$-flexibility: every set of at most $k$ prescribed edges can remain uncrossed in some 1-planar drawing. We use this property to reconstruct 1-planar drawings of larger graphs from drawings of smaller $k$-flexible graphs. This replaces exhaustive testing of more than forty billion cubic graphs with computations on far fewer graphs of smaller order.

The Minimum Sum Vertex Cover (MSVC) problem asks for an ordering of the vertices of a graph that minimizes the sum, over all edges, of the time at which each edge is first covered. We study the problem through the structure of vertex covers and obtain new approximation and exact algorithms, together with conditional lower bounds. For graphs of maximum degree $\Delta$, we show that a simple ordering algorithm based on a minimum vertex cover achieves approximation ratio $R_\Delta \le {(\sqrt{\Delta}+1)}/{2}$. For $d$-regular graphs, we give a polynomial-time $1.184$-approximation by combining Max-$k$-Vertex-Cover approximation with a structural bound on optimal prefixes. On the exact side, we give an algorithm parameterized by the vertex cover number $k$ running in $2^{O(k\log k)} + O(n+m)$ time, improving the previous dependence on $k$, where $n$ and $m$ are the number of vertices and edges in the graph, respectively. We also develop a separator-based exact algorithm running in $ 2^{O(\sqrt n \log n)}$ time on planar, bounded-genus, and fixed-minor-free graph classes. Finally, we prove that Minimum Sum Vertex Cover is NP-hard on planar graphs and, assuming ETH, admits no $2^{o(\sqrt n)}$-time exact algorithm on $n$-vertex planar graphs. Thus our planar upper bound is tight up to logarithmic factors in the exponent.

In the chip testing problem, we are given $n$ chips, strictly more than half of which are good. Chips can test one another in pairs; a good chip always reports the status of the other chip correctly, whereas a bad chip may report arbitrarily and adversarially. The goal is to identify a single chip that is guaranteed to be good. The problem originates in system-level fault diagnosis and is closely related to the "knights and spies" puzzle. The standard textbook solution is a halving recursion that tests disjoint pairs in rounds and keeps one chip from each consistent pair. We present the Backtracking Candidate Elimination (BCE) algorithm, a sequential alternative that scans the chips once while maintaining a current candidate and a stack of retained chips. Every chip is tested at most once as the incoming chip; when a test is inconclusive the candidate and the incoming chip are discarded together, and the algorithm backtracks to the most recently retained chip. BCE uses at most $n-1$ tests and $O(n)$ time, needs no parity case analysis, and works online. Its correctness follows from two invariants: the retained chips all have the same type, and every discarded pair contains at least one bad chip. We explain how BCE can be viewed as the Boyer-Moore majority vote algorithm with its counter replaced by a stack of physical witnesses, and why that replacement is needed. We also give an early termination rule and a variant for the weaker model of one-directional tests.

We study representations for relation algebras corresponding to certain edge colourings of complete graphs. Previously suitable colourings were obtained for the number of colours $n$ up to $2000$, with two exceptions: $n = 8$ and $n = 13$. Using a method that we have called the FBF (Fusion Beautiful Fusions) method, we find colourings for $8$ and $13$ colours. Our method is a new guess-and-check approach that we describe as an adaptation of Comer's finite-field method. Using FBF we construct novel colourings that are non-isomorphic to existing published colourings for 5, 6, 7, 9, 10, 11 and 12 colours.

A length-$n$ binary $k$-deletion code is a set of binary strings such that if we delete any $k$ bits of a string, leaving a length-$(n-k)$ binary string, we can uniquely recover the codeword. In this paper, we consider $t$-list decodable deletion codes, where after $k$ bits of a codeword are deleted, we can identify a list of size at most $t$ such that the original codeword lies in the list. We prove a lower bound of $\Omega_k(2^n t\log^{1/t}n/n^{k+k/t})$ on the optimal size of a $t$-list decodable $k$-deletion code, giving a $\sqrt{\log n}$ improvement over the previously best known bounds for $2$-list decodable $2$-deletion codes [GH21] and providing the first nontrivial lower bound when $t>2$ or $k>2$. Our bound holds for all $t\leq n^k$, showing that $t=\Omega(\log n)-$list decodable deletion codes have optimal size $\Theta_k(2^n t/n^k)$, asymptotically matching the known upper bound. We also prove upper bounds on the number of common subsequences and common supersequences of a given length for any two binary strings.

Let $f\in\mathbb F_q[x]$ be a monic polynomial of degree $m$ with $f(0)\ne 0$, and let $\mathcal R=\mathbb F_q[x]/\langle f\rangle$. Under coefficient expansion, a quasi-polycyclic (QP) code of index $n$ corresponds to an $\mathcal R$-submodule of $\mathcal R^n$. In this paper, we study QP codes with respect to the annihilator duality. We show that this form is non-degenerate and that the annihilator dual of a QP code is again a QP code. We also give an equivalent description of the dual in terms of the $\mathcal R$-valued dot product, which leads to self-orthogonality criteria. We determine the Gram matrix of the annihilator form and obtain an explicit formula for its determinant. In coefficient coordinates, this shows that the annihilator dual can be viewed as a twisted Euclidean dual. Using this description, we characterize when a coordinatewise $\mathbb F_q$-linear map converts annihilator duality into ordinary Euclidean duality. For squarefree $f$, we show that annihilator duality decomposes into ordinary Euclidean duality on the components arising from the Chinese Remainder Theorem. This gives simple criteria for self-orthogonal, self-dual, dual-containing, and complementary-dual QP codes. We show how the annihilator dual interacts with the Hamming weight enumerator and compute the MacWilliams transform associated with that duality. Finally, we apply these results to Calderbank–Shor–Steane and Steane-enlarged quantum-code constructions over $\mathcal R$ and, when a suitable duality-preserving coordinate map exists, over $\mathbb F_q$. This gives binary and ternary stabilizer codes with minimum-distance lower bounds matching the best known bounds, most of which arise from rings $\mathcal R$ that are not fields.

We exhibit simple, rotationally symmetric Venn diagrams with 17 curves, with 19 curves and with 23 curves: $n$ Jordan curves carried to one another by rotation through $2\pi/n$, with every one of the $2^n$ regions present and connected and, since the diagrams are simple, every crossing on exactly two curves. Symmetric Venn diagrams exist for every prime number of curves (Griggs, Killian and Savage, 2004), but those diagrams have many curves through a point; simple ones were known only up to 13 curves (Mamakani and Ruskey, 2014). Four 17-curve, twelve 19-curve and six 23-curve diagrams were found by a Metropolis walk on rotation-invariant quadrangulations of the sphere in which regions may temporarily be duplicated, started from the Griggs-Killian-Savage diagram with its multiple crossings resolved. At 23 curves the walk was held for two weeks by duplicated regions near the poles; the two lineages that finished were the first whose $E=92$ states had none. Every diagram is given by a machine-checkable certificate; one certificate each of the 17-, 19- and 23-curve sizes has been verified by a formal proof in Lean 4. All of the diagrams are non-monotone, which is why the crossing-sequence searches that found the 11- and 13-curve diagrams could not have found them.

Inspired by the static problem $(\alpha,\beta)$-Dominating Set, we propose a general temporal domination problem, called $(\alpha,\beta)$-Temporal Dominating Set ($(\alpha,\beta)$-TDS). We show that this problem encompasses Temporal Dominating Set, and additionally provides first temporal extensions of problems such as $k$-Dominating Set and $\alpha$-Dominating Set. In this paper, we study the parameterised complexity of $(\alpha,\beta)$-TDS with respect to temporal neighbourhood diversity (TND), temporal modular-width (TMW), and temporal cliquewidth (TCW). We obtain fixed parameter tractability results for all values of $\alpha$ and $\beta$ with respect to TND; W[1]-hardness with respect to TMW and TCW whenever $\beta$ is in the problem input, or whenever $\alpha\in (0,1)$ and $\beta$ is a fixed constant; and para-NP-hardness with respect to TCW when $\alpha = 0$ and $\beta = 1$, or $\alpha = 1$ and $\beta = 0$.

We consider the problem of determining if a given two-dimensional nonnegative integer matrix $M$ is the product of two such matrices, excluding trivial units. A matrix $M$ with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of $2 \times 2$ matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

All the eigenvalues of an integral graphs are integers. Integral graphs are extremely rare. They form an asymptotically vanishing fraction $2^{-\Omega(n)}$ among all graphs on $n$ vertices. It makes the construction of a new family of integral graphs a challenging task. Also, most of the known infinite family of integral graphs rely on Cayley graphs over Abelian groups. In this article, we introduce a new family of integral graphs obtained from the groups. The construction of our graphs from groups is different from the construction of Cayley graphs. A spectral uniqueness theorem is established, which shows that each member of the infinite family is determined by its adjacency spectrum among all finite simple graphs. We also present recursive constructions that generates larger members of the family from smaller ones, providing a scalable class of integral graphs. Finally, we investigate algebraic realizations of these graphs as complements of Proper Prime Order Element Graphs of finite $2$-groups and obtain conditions characterizing such realizations. We also observe that the graphs obtained from different non-isomorphic groups have cospectral graphs.

We construct an explicit quaternary Legendre pair of length $64$. To the best of our knowledge, this is the first reported construction at this length.

In 1984, Frank Harary introduced the first convexity games in graphs, all of them based on the geodesic convexity, which is the graph convexity related to shortest paths. In 2024, Araújo et al. obtained the first PSPACE-hardness proofs on some of these geodesic games and generalized them to any graph convexity. In this paper, we investigate convexity games on several known induced path convexities: the monophonic $\mathrm{m}$-convexity and the $\ell_k$-convexities, based on induced paths and on induced paths of size at most $k$. We prove that the hull games $\mathrm{CHG}_{\mathrm{m}}$, $\mathrm{CHG}_{\ell_k}$ and their misère variants are PSPACE-complete for every $k\ge2$ even in graphs with diameter at most 3. We also use the Sprague-Grundy Theory to obtain a polynomial time algorithm to decide the winner of the games $\mathrm{CHG}_{\mathrm{m}}$ and $\mathrm{CHG}_{\ell_k}$ for any $k\ge2$ in disjoint unions of paths and cycles. For $k\ge3$ odd, we prove that Alice (1st player) wins $\mathrm{CHG}_{\ell_k}$ in the path $P_n$ if and only if $n$ is odd and she wins in the cycle $C_n$ if and only if $n=3$ or $n=\alpha\cdot (k+1)-1$ with $\alpha\ge2$. For $k\ge2$ even, the only periodic nimber sequences of $\mathrm{CHG}_{\ell_k}$ obtained through extensive computational testing occurred for $k=2^h-4$ with $h\ge3$, e.g, $k\in\{4,12,28,60,\ldots\}$. In this case ($k=2^h-4$ with $h\ge3$), we prove that the nimber sequences of $\mathrm{CHG}_{\ell_k}$ in $P_n$ and in $C_n$ are periodic and Alice loses (resp. wins) in $P_n$ (resp. $C_n$) with $n>k$ only when $n=3k+4$ (resp. $n\in\{2k+1,5k+3\}$). Finally, we show that, for $k=2$, the game $\mathrm{CHG}_{\ell_2}$ in paths $P_n$ is closely related to the classical game Couples-are-Forever of J. H. Conway: it is still an open problem if the nimber sequence is periodic or not and Alice loses only for 12 values of $n$ up to $50$ million.

We determine the asymmetric homomorphism threshold from graphs of odd girth at least $7$ to triangle-free graphs, showing that $\delta_{\mathrm{hom}}(\{C_3,C_5\},\{C_3\})=\frac{1}{9}$. Equivalently, for every $\varepsilon>0$, every $n$-vertex graph of odd girth at least $7$ and minimum degree at least $(1/9+\varepsilon)n$ admits a homomorphism to a triangle-free graph of size bounded by a function of $\varepsilon$, while there exist graphs of odd girth at least $7$ and minimum degree at least $(1/9-\varepsilon)n$ for which no such bounded-size triangle-free homomorphic image exists. More generally, for every $t\geq 3$, we prove $\delta_{\mathrm{hom}}(\{C_3,C_5\},\{K_t\})=\frac{1}{3t}$. In particular, for every proper monotone class $\mathcal C$ of graphs, the threshold for graphs of odd girth at least $7$ to admit a homomorphism to a bounded-size graph in $\mathcal C$ is positive. We further extend this phenomenon to arbitrary odd girth: for every $k\geq 2$ and every proper monotone subclass $\mathcal C$ of the class of graphs of odd girth at least $2k-1$, the threshold for graphs of odd girth at least $2k+3$ to admit a homomorphism to a bounded-size graph in $\mathcal C$ is positive. These results disprove conjectures of Gishboliner, Hurley and Wigderson and exhibit a sharp contrast with the corresponding zero chromatic-threshold results. Our lower-bound constructions are based on high-dimensional Borsuk graphs, while the matching upper bounds use regularity arguments to recover the structure underlying these constructions.

In the online metric matching problem, we have $n$ servers with known locations in some metric space. Requests arrive one-by-one at certain locations, and upon arrival a request must be matched to a server that was not matched to a previous request. The goal is to minimize the matching cost. For randomized algorithms with an oblivious adversary, the best known competitive ratio is obtained by embedding the metric space into an HST, and then solving the problem in the setting where the metric space is defined by the HST. Bansal et al. (Algorithmica, 2014) introduced a framework for online metric matching where one develops an algorithm in a restricted reassignment model, and then transforms this into a true online algorithm. Using this framework, they obtained an expected competitive ratio of $O(\log n)$ for HSTs; this also gives the best known competitive ratio of $O(\log^2 n)$ for general metrics. In this paper, we revisit this framework with the aim of developing new algorithms. For HSTs where each node has at most $\Delta$ children, we develop an algorithm via this framework with an expected competitive ratio of $O((\log\log \Delta) \cdot \log \Delta)$. In particular, this ratio is independent of $n$, the number of servers/requests. It is near-optimal, as the expected competitive ratio of any algorithm is $\Omega(\log \Delta)$.

We consider the offline problem of parallel relocation of k identical mobile resources. After each request, known in advance, the resources may move simultaneously, and the duration of a step is determined by the longest individual movement. This model is equivalent to the offline version of the time-optimal k-server problem. We prove that the decision version is strongly NP-complete already on metrics of finite subsets of the Euclidean line, or equivalently, on vertex metrics of weighted paths. Thus, the computational hardness persists even under a linear arrangement of the admissible resource locations. As a positive result, we show that the problem can be solved exactly in polynomial time on metrics of undirected unweighted graphs with a universal vertex.

We show that isomorphism of $F$-free tournaments can be solved in FPT time $f(k) \cdot n^{O(1)}$, where $k$ denotes the size of $F$, and $n$ denotes the size of the input tournaments. Our result extends on a previous FPT isomorphism test for tournaments of bounded twin-width [Grohe, Neuen 2024], as well as XP isomorphism tests parameterized by the VC dimension or the chromatic number [Raßmann, Schweitzer 2026]. It also implies that every non-trivial hereditary class of tournaments admits a polynomial-time isomorphism test. Our algorithm builds on a novel combination of spectral, geometric, combinatorial and group-theoretic tools.

We provide a clearer presentation of the Chair44 monotile, a new strongly aperiodic monotile in three-dimensional Euclidean space $E^3$.

Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions. These inequalities have been established primarily for the uniform measure and for biased product measures, often through Fourier-analytic or inductive arguments. We develop a curvature-based semigroup framework to establish isoperimetric inequalities for measures with weakly dependent coordinates. In particular, we show that a Dobrushin-type condition, together with marginal boundedness of the distribution, implies the local Bobkov inequality, Talagrand's $L^1$–$L^2$ and variance–surface-area inequalities, the Kahn–Kalai–Linial inequality, and the Eldan–Gross inequality. Our results apply to zero-field Ising models with interaction matrix $J$ throughout the Dobrushin uniqueness regime $\|J\|_1<1$. The framework builds on discrete Bakry–Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.

In the classical Art Gallery Problem (AGP), guards are placed in a polygon so that together they see every point. The Witness Set Problem (WSP), introduced by Amit, Mitchell, and Packer, is a natural dual to the AGP. In this paper, we study the WSP in weak visibility polygons (WVPs), the simple polygons in which every point is seen from some point of one fixed edge. A witness set is a set of points whose visibility regions are pairwise disjoint, so that no single guard sees two of them. A maximum witness set, therefore, lower-bounds the guard number. Exact polynomial-time algorithms for the WSP are known only for monotone mountains, a proper subclass of WVPs. We give the first exact polynomial-time algorithms for the WSP in WVPs, in two settings. In the Discrete Witness Set Problem (DiscWSP), the witnesses come from a given set of $m$ points, and we find a maximum witness subset in $O(n + m \log(n+m))$ time on an $n$-vertex polygon. The algorithm rests on a structural fact: the visibility intersection graph of a WVP, in which two points are adjacent if their visibility regions intersect, is a trapezoid graph, that is, an intersection graph of trapezoids between two parallel lines. Moreover, the class of these graphs properly contains the interval graphs and the permutation graphs, which may be of independent interest in graph theory. We also prove an $\Omega(n \log n)$ lower bound in the algebraic decision-tree model for instances with $m = \Theta(n)$, so our algorithm for DiscWSP is optimum. In the Continuous Witness Set Problem (ContWSP), a witness may be any point of the polygon, and we give an exact algorithm running in $O(n \log n + \rho^{2}(n + \rho^{2}))$ time, where $\rho$ is the number of reflex vertices.

A majority coloring of a graph $G$ is a vertex coloring of $G$ in which no vertex has more than half of its neighbors colored with its own color. The least number of colors required for a majority coloring of $G$ is the majority chromatic number $\mu(G)$. The majority coloring game, introduced by Bosek et al. (2019), is a two-player Maker-Breaker-type game where the players alternately color vertices while maintaining the majority condition at each vertex; the least number of colors required for the first player to have a winning strategy on $G$ is the majority game chromatic number $\mu_g(G)$. In contrast with the static case, Bosek et al. show that $\mu_g(G)$ is unbounded in general, while $\mu_g(G)\le col_g(G)$, where $col_g(G)$ is the game coloring number of $G$. It is known (cf. Faigle et al. (1993)) that, for any acyclic graph $G$, $col_g(G) \leq 4$, so $\mu_g(G) \leq 4$ as well. We improve this bound by showing that $\mu_g(G) \leq 3$ for any acyclic graph $G$ of maximum degree at most $4$. We also show that $\mu_g(G) = 2$ if $G$ is a nonempty path, star, or complete graph, or a disjoint union of such graphs. These improve the results of Bosek et al. We also initiate the study of the computational complexity of the pre-coloring extension problem for the static and game versions of majority coloring. We show that the Majority $k$-precoloring Extension problem is NP-complete for each $k \geq 2$. When $k = 2$, it remains NP-complete on the class of bipartite graphs where maximum degrees of each part are $5$ and $6$, and when $k = 3$ it remains NP-complete on the class of $12$-regular planar graphs. We show that the game version is PSPACE-complete, even when the number of colors $k=\chi(G)$, and even when $k=2$ and $G$ is a bipartite graph in which one part has maximum degree equal to $9$. The results on the game versions also hold for the variation where Bob starts the game.

Let $\sigma_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. We completely determine the sensitivity, average sensitivity, and block sensitivity of $\sigma_{n,d}$ for every $1\le d\le n$. Using Lucas' theorem, we obtain a uniform binary description of the Hamming-weight value sequence, from which the sensitivity and average-sensitivity formulas follow and the computation of block sensitivity reduces to at most four explicit candidates. Combining these results with the arbitrary-degree formula for certificate complexity, we determine the exact relations among sensitivity, block sensitivity, and certificate complexity. We also prove a general result for symmetric Boolean functions: every nonconstant symmetric Boolean function $f$ satisfies \[ \bs(f)\le \max\{s(f),C(f)-1\}. \] Consequently, only the three patterns \[ s=\bs=C,\qquad s=\bs<C,\qquad s<\bs<C \] can occur for nonconstant symmetric Boolean functions. For elementary symmetric Boolean functions, we give necessary and sufficient conditions for each of these three patterns, thereby completely classifying the relations among $s(\sigma_{n,d})$, $\bs(\sigma_{n,d})$, and $C(\sigma_{n,d})$. In particular, we obtain a necessary and sufficient characterization of the full strict hierarchy \[ s(\sigma_{n,d})<\bs(\sigma_{n,d})<C(\sigma_{n,d}), \] and exhibit infinite families for which it holds.

We show that Arc Kayles is PSPACE-complete. This solves a question raised by Schaefer in 1978.

For a fixed integer $r\geq4$, the $K_r-e$-free graph sandwich problem asks whether, given graphs $G_1\subseteq G_2$ on the same vertex set, there is an induced-$K_r-e$-free graph $H$ between them. We give a deterministic algorithm taking $O(n+\alpha(G_2)^{r-3}m_2)$ time and space, where $m_2=|E(G_2)|$ and $\alpha(G_2)$ is the arboricity of $G_2$. In particular, the diamond-free case takes $O(n+\alpha(G_2)m_2)$ time. This improves the direct $O(n^r m_2)$ implementation of the previously known forced-edge closure. Our implementation maintains components of common neighborhoods indexed by $(r-3)$-cliques. A filtered frontier supports their merges within the clique-listing bound, while completion events avoid repeatedly searching for affected cliques. On feasible instances the output is contained in every feasible sandwich, independently of processing order. Applying the closure to $(G,K_n)$ gives an $O(n^{r-1})$-time bound for partitioned and nonpartitioned probe $K_r-e$-free recognition, improving the $O(n^{r+2})$ bound obtained from the direct sandwich closure. We also describe a direct static recognizer based on the same local characterization.

For integers $1\le k\le n/2$, let $f(k,n)$ be the least integer $s$ such that every $s$-connected graph on $n$ vertices contains a spanning bipartite $k$-connected subgraph. Thomassen conjectured that $f(k,n)$ is bounded by a function of $k$ alone. Delcourt and Ferber proved $f(k,n)=O(k^3\log n)$, and Yuster subsequently obtained $f(k,n)\le22k^2\log_2 n$. We prove that, for $2\le k\le n/2$, \[ f(k,n)\le\min\left\{n-1,\, \left\lfloor6(k-1)\log_2\frac{n}{k-1}\right\rfloor\right\}. \] In particular, $f(k,n)=O(k\log(n/k))$.

We study the number of prototypes needed to represent Boolean functions by nearest-neighbour classification. There are two distinct settings: the prototypes may be arbitrary points of Euclidean space, or they may themselves be required to lie in the Boolean cube. For unrestricted prototypes, we strengthen a known lower bound for almost all Boolean functions. The bound applies simultaneously to nearest-neighbour voting rules with any number of voting neighbours, and substantially narrows the gap with the known general upper bound. We obtain a VC-dimension bound for classes with a bounded number of prototypes, and show that it is sharp in order in dimensions at least four. We then study Boolean prototypes, beginning with symmetric threshold functions. A connection with covering designs expresses the minimum number of prototypes at every threshold level exactly in terms of a covering number, and leads to further exact results for related monotone functions, including disjunctive extensions and a characterisation of when a representation with a single negative prototype is possible. For a uniformly random Boolean function, the Boolean nearest-neighbour complexity, as a proportion of the cube, is asymptotically close either to one half or to one, with explicit limiting probabilities. In particular, almost every Boolean function requires at least approximately half as many prototypes as there are points in the cube, and one half is the largest proportion for which such a lower bound holds. Finally, we consider arbitrary symmetric Boolean functions. Their Boolean nearest-neighbour complexity is closely approximated by a weighted vertex-cover problem on paths. As a consequence, a uniformly random symmetric function typically requires prototypes amounting to $11/20$ of the cube. This is much larger than the upper bounds known when the prototypes are allowed to lie anywhere in Euclidean space.

Given a directed graph with positive edge weights and two vertices s,t, a next-to-shortest s-t path is a shortest simple s-t path among those whose length is strictly larger than the shortest-path distance. The problem was introduced by Lalgudi, Papaefthymiou and Potkonjak in 1996; it is NP-hard when zero-weight edges are allowed, and its complexity on positively weighted digraphs remained open for almost three decades until Chen, Wein and Zhang recently gave a polynomial-time algorithm running in O(n^4 m^3 log n) time. We give a substantially faster algorithm within their optimal-middle-segment framework. The core idea is to split the problem into "choosing a prefix" and "completing it". Given a prefix P: s -> A made of shortest-path edges, delete the vertices used by P, forbid leaving A along shortest-path edges, and the best completion is one shortest-path computation. The difficulty lies in choosing P: even for a fixed A, deciding whether some shortest prefix admits a completion is NP-complete. We do not solve these fixed-A subproblems one by one. Fix any globally optimal next-to-shortest path; its middle segment induces a boundary edge x -> c in the shortest-path DAG. For the correct triple (A,B,x), the optimal path certifies c as a feasible next hop, and we prove that every feasible next hop that is not earlier than c in a topological order can be combined with the same middle segment into another globally optimal path. Hence only the feasible next hop of maximum topological index is kept per triple, giving O(n^3) representatives, all generated by a two-dimensional DAG dynamic program with a local reward. The total running time is O(n^3 (m + n log n)), and O(n^3 m) on unweighted graphs. The proof rests on an uncrossing lemma: the last intersection between a reference prefix and the candidate's partner suffix can always be moved strictly earlier, which cannot go on forever.