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A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lovász proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple edges. Denote by b(G) the number of bricks in such a decomposition. An edge e of G is removable if G-e is also matching covered; is b-invariant if e is removable and b(G-e)=b(G). Furthermore, an edge e of G is a forcing edge if it lies in precisely one perfect matching of G. Lucchesi and Murty proposed the problem of characterizing bricks, distinct from K_4, \overline{C_6}, and the Petersen graph, in which every b-invariant edge is a forcing edge. In this paper, we solve this problem for near-bipartite bricks by providing a complete characterization.

Let $A$ be a finite dimensional algebra over an algebraically closed field. Motivated by some foundational interactions between bricks and $τ$-rigid modules, we prove, in full generality, that if all but finitely many bricks of the algebra $A$ are $τ$-rigid, then $A$ is brick-finite. Equivalently, any brick-infinite algebra admits infinitely many bricks which are not $τ$-rigid. Because $τ$-rigidity implies rigidity, our result verifies a weaker version of an open conjecture which states that if (almost) all bricks over $A$ are rigid, then $A$ should be brick-finite. In retrospect, this work strengthens some previous results and contributes to the recent studies of a series of challenging problems, all tied to the $2$nd brick-Brauer-Thrall conjecture. More specifically, without any tameness assumption, we settle a question that was previously known only for $E$-tame algebras.

General-purpose programmable photonic processors provide a flexible foundation for integrating various functionalities within a single chip. A two-dimensional bricks waveguide mesh of Mach Zehnder interferometers has been demonstrated to possess considerable potential in the domain of photonic neural networks and quantum signal processing. In this article, we propose an expansion of the available applications of recirculating bricks mesh architecture to distillation protocols necessary for quantum signal processing. These protocols are essential for the heralding of the output of single photons, which is characterized by a reduced distinguishability error rate. The demonstration will be made of a single programmable optical system's ability to realize various distillation protocols with reduced computational resource costs. The present study will concentrate on cascaded quantum interferometers and Fourier transform-based schemes. It will demonstrate that the bricks mesh can implement such schemes, which are unattainable using feed-forward networks, without the need for complex out-of-plane integration. The propagation of the signal in any direction, along with the utilization of al

A connected graph G is matching covered if every edge lies in some perfect matching of G. Lovasz proved that every matching covered graph G can be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite) up to multiple edges. Denote by b(G) the number of bricks of G. An edge e of G is removable if G-e is also matching covered, and solitary (or forcing) if after the removal of the two end vertices of e, the left graph has a unique perfect matching. Furthermore, a removable edge e of a brick G is b-invariant if b(G-e) = 1. Lucchesi and Murty proposed a problem of characterizing bricks, distinct from K4, the prism and the Petersen graph, in which every b-invariant edge is forcing. We answer the problem for cubic bricks by showing that there are exactly ten cubic bricks, including K4, the prism and the Petersen graph, every b-invariant edge of which is forcing.

We dream of AI agents that can read arbitrary designs and construct real-world objects from reusable building blocks. As a first step toward this vision, we study whether multimodal large language models (MLLMs) possess the visual grounding and spatial reasoning capabilities required for brick assembly. We formulate brick assembly as a sequential decision-making problem, where each step involves two subtasks: brick selection, identifying the target brick from candidate components, and brick pose estimation, predicting where and how the selected brick should be placed. To support this study, we introduce BC-Bench (Brick Construction Benchmark), the first benchmark for evaluating MLLMs on assembly with diverse bricks. Experiments show that current state-of-the-art MLLMs remain far from reliable builders, struggling with fine-grained brick selection and failing at precise pose estimation. To bridge this gap, we propose Brick-Composer, a learning framework that equips MLLMs with assembly skills through three complementary signals: Human Design Sparks, which provide affordance-rich construction demonstrations; World Feedback, which grounds predicted actions in visual and physical conseq

Diffusion models excel at generating photo-realistic images but come with significant computational costs in both training and sampling. While various techniques address these computational challenges, a less-explored issue is designing an efficient and adaptable network backbone for iterative refinement. Current options like U-Net and Vision Transformer often rely on resource-intensive deep networks and lack the flexibility needed for generating images at variable resolutions or with a smaller network than used in training. This study introduces LEGO bricks, which seamlessly integrate Local-feature Enrichment and Global-content Orchestration. These bricks can be stacked to create a test-time reconfigurable diffusion backbone, allowing selective skipping of bricks to reduce sampling costs and generate higher-resolution images than the training data. LEGO bricks enrich local regions with an MLP and transform them using a Transformer block while maintaining a consistent full-resolution image across all bricks. Experimental results demonstrate that LEGO bricks enhance training efficiency, expedite convergence, and facilitate variable-resolution image generation while maintaining stron

A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. Lucchesi and Murty proposed a problem concerning the characterization of bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. In this paper, we present a characterization of this problem when the bricks are claw-free.

An edge of a matching covered graph G is a forcing edge if it lies in precisely one perfect matching of G. A matching covered graph is a brick if and only if it is 3-connected and bicritical (the deletion of each pair of distinct vertices results in a graph with a perfect matching). In this paper, we prove that every vertex of a brick is incident with a forcing edge if and only if the brick is an odd wheel up to multiple edges.

A graph $G$ is a brick if it is 3-connected and $G-\{u,v\}$ has a perfect matching for any two distinct vertices $u$ and $v$ of $G$. A brick $G$ is solid if for any two vertex disjoint odd cycles $C_1$ and $C_2$ of $G$, $G-(V(C_1)\cup V(C_2))$ has no perfect matching. Lucchesi and Murty proposed a problem concerning the characterization of bricks, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, in which every $b$-invariant edge is solitary. In this paper, we show that for a solid brick $G$ of order $n$ that is distinct from $K_4$, every $b$-invariant edge of $G$ is solitary if and only if $G$ is a wheel $W_n$.

We study infinite string modules that are bricks over some gentle algebras. In particular, we first give a complete classification of these modules over the double-Kronecker gentle algebra and prove that each family is in bijection with a family of Sturmian (binary) words. We then generalize some of our results to a larger family of gentle algebras.

Motivated by the recent work of Deaconu, Mousavand and Paquette on the connection between infinite string bricks for certain gentle algebras and Sturmian words, we develop a decorated version of a deterministic automaton, called a multi-entry inverse automaton (MIA, for short) that accepts pointed words. We then associate an MIA $\mathsf M_{Λδ}$ over $\{0,1\}$ to a string algebra $Λ$, and show that strings over $Λ$ can be viewed as certain equivalence classes of the pointed words accepted by $\mathsf M_{Λδ}$. By defining (weak) brick words over this MIA, we show that a finite/infinite string module (resp. band module) is a brick if and only if every word in the associated equivalence class of pointed binary words is a brick word (resp. a weak brick word) over $\mathsf M_{Λδ}$. The result of Deaconu et al. follows as an immediate consequence.

In the standard DNA brick set-up, distinct 32-nucleotide strands of single-stranded DNA are each designed to bind specifically to four other such molecules. Experimentally, it has been demonstrated that the overall yield is increased if certain bricks which occur on the outer faces of target structures are merged with adjacent bricks. However, it is not well understood by what mechanism such `boundary bricks' increase the yield, as they likely influence both the nucleation process and the final stability of the target structure. Here, we use Monte Carlo simulations with a patchy particle model of DNA bricks to investigate the role of boundary bricks in the self-assembly of complex multicomponent target structures. We demonstrate that boundary bricks lower the free-energy barrier to nucleation and that boundary bricks on edges stabilize the final structure. However, boundary bricks are also more prone to aggregation, as they can stabilize partially assembled intermediates. We explore some design strategies that permit us to benefit from the stabilizing role of boundary bricks whilst minimizing their ability to hinder assembly; in particular, we show that maximizing the total number

Generalising a recent work of Dequêne et al. on the connection between perfectly clustering words and band bricks over a particular family of gentle algebras, we characterise band bricks over string algebras whose underlying quiver is acyclic in terms of weakly perfectly clustering pairs of words -- a variant of perfectly clustering words. As a consequence, we characterise band semibricks over all such algebras. Furthermore, the combination of our result and a result of Mousavand and Paquette provides an algorithm to determine whether such a string algebra is brick-infinite.

In this paper, a system to build music in an intuitive and accessible way, with Lego bricks, is presented. The system makes use of the new powerful and cheap possibilities that technology offers for making old things in a new way. The Raspberry Pi is used to control the system and run the necessary algorithms, customized Lego bricks are used for building melodies, custom electronic designs, software pieces and 3D printed parts complete the items employed. The system designed is modular, it allows creating melodies with chords and percussion or just melodies or perform as a beatbox or a melody box. The main interaction with the system is made using Lego-type building blocks. Tests have demonstrated its versatility and ease of use, as well as its usefulness in music learning for both children and adults.

A semibrick is a set of modules satisfying Schur's Lemma, and it is said to be maximal if it is not properly contained in another semibrick. For any finite dimensional algebra $\varLambda$ over an algebracally closed field $K$, we prove that any maximal finite semibrick $\mathcal{S}$ consists only of open bricks $B$, that is, bricks whose orbit closures $\overline{\mathcal{O}_B}$ are irreducible components in the representation schemes.

We prove that if $Λ$ is a tame finite-dimensional algebra over an algebraically closed field and $G$ is a generic $Λ-$module, then $G$ is a generic brick if and only if it determines a one-parameter family of bricks with the same dimension. In particular, we obtain that $Λ$ admits a generic brick if and only if $Λ$ is brick-continuous.

A connected graph G with at least two vertices is matching covered if each of its edges lies in a perfect matching. We say that an edge e in a matching covered graph G is removable if G-e is matching covered. A pair {e; f} of edges of a matching covered graph G is a removable doubleton if G-e-f is matching covered, but neither G-e nor G-f is. Removable edges and removable doubletons are called removable classes, introduced by Lovasz and Plummer in connection with ear decompositions of matching covered graphs. A 3-connected graph is a brick if the removal of any two distinct vertices, the left graph has a perfect matching. A brick G is wheel-like if G has a vertex h, such that every removable class of G has an edge incident with h. Lucchesi and Murty proposed a problem of characterizing wheel-like bricks. We show that every wheel-like brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels in a certain manner. A matching covered graph is minimal if the removal of any edge, the left graph is not matching covered. Lovasz and Plummer proved that the minimum degree of a minimal matching covered bipartite graph different from K2 is 2 by ear decompositions i

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. A nonbipartite matching covered graph $G$ is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi, and Murty proved that every brick other than $K_4$ and $\overline{C_6}$ has at least $Δ-2$ removable edges. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1,e_2\}$ such that $G-\{e_1,e_2\}$ is a bipartite matching covered graph. In this paper, we show that in a near-bipartite brick $G$ with at least six vertices, every vertex of $G$, except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, $G$ has at least $\frac{|V(G)|-6}{2}$ removable edges. Moreover, all graphs attaining this lower bound are characterized.

The bricks over preprojective algebras of type A are known to be in bijection with certain combinatorial objects called "arcs". In this paper, we show how one can use arcs to compute bases for the Hom-spaces and first extension spaces between bricks. We then use this description to classify the "weak exceptional sequences" over these algebras. Finally, we explain how our result relates to a similar combinatorial model for the exceptional sequences over hereditary algebras of type A.