Edge Contraction Research Articles

New Constructions Related to the Polynomial Sphere Recognition Problem

We construct a simply connected 2-complex C embeddable in 3-space such that for any embedding of C in $${\mathbb {S}}^3$$ , any edge contraction forms a minor of the 2-complex not embeddable in 3-space. Additionally, we construct the 2-complex C such that every edge of C forms a nontrivial knot in any of the embeddings of C in $${\mathbb {S}}^3$$ . This answers a question of Saul Schleimer.

Making graphs compact by lossless contraction

This paper proposes a scheme to reduce big graphs to small graphs. It contracts obsolete parts and regular structures into supernodes. The supernodes carry a synopsis S_mathcal {Q} for each query class mathcal {Q} in use, to abstract key features of the contracted parts for answering queries of mathcal {Q}. Moreover, for various types of graphs, we identify regular structures to contract. The contraction scheme provides a compact graph representation and prioritizes up-to-date data. Better still, it is generic and lossless. We show that the same contracted graph is able to support multiple query classes at the same time, no matter whether their queries are label based or not, local or non-local. Moreover, existing algorithms for these queries can be readily adapted to compute exact answers by using the synopses when possible and decontracting the supernodes only when necessary. As a proof of concept, we show how to adapt existing algorithms for subgraph isomorphism, triangle counting, shortest distance, connected component and clique decision to contracted graphs. We also provide a bounded incremental contraction algorithm in response to updates, such that its cost is determined by the size of areas affected by the updates alone, not by the entire graphs. We experimentally verify that on average, the contraction scheme reduces graphs by 71.9% and improves the evaluation of these queries by 1.69, 1.44, 1.47, 2.24 and 1.37 times, respectively.

Multiple Superpixel Graphs Learning Based on Adaptive Multiscale Segmentation for Hyperspectral Image Classification

Hyperspectral image classification (HSIC) methods usually require more training samples for better classification performance. However, a large number of labeled samples are difficult to obtain because it is cost- and time-consuming to label an HSI in a pixel-wise way. Therefore, how to overcome the problem of insufficient accuracy and stability under the condition of small labeled training sample size (SLTSS) is still a challenge for HSIC. In this paper, we proposed a novel multiple superpixel graphs learning method based on adaptive multiscale segmentation (MSGLAMS) for HSI classification to address this problem. First, the multiscale-superpixel-based framework can reduce the adverse effect of improper selection of a superpixel segmentation scale on the classification accuracy while saving the cost to manually seek a suitable segmentation scale. To make full use of the superpixel-level spatial information of different segmentation scales, a novel two-steps multiscale selection strategy is designed to adaptively select a group of complementary scales (multiscale). To fix the bias and instability of a single model, multiple superpixel-based graphical models obatined by constructing superpixel contracted graph of fusion scales are developed to jointly predict the final results via a pixel-level fusion strategy. Experimental results show that the proposed MSGLAMS has better performance when compared with other state-of-the-art algorithms. Specifically, its overall accuracy achieves 94.312%, 99.217%, 98.373% and 92.693% on Indian Pines, Salinas and University of Pavia, and the more challenging dataset Houston2013, respectively.

On the Parameterized Complexity of Maximum Degree Contraction Problem

In the Maximum Degree Contraction problem, the input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time $$n^{\mathcal {O}(k)}$$ . As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van’t Hof, and Paulusma studied the problem in the realm of parameterized complexity and proved, among other things, that it admits an FPT algorithm running in time $$(d + k)^{2k} \cdot n^{\mathcal {O}(1)} = 2^{\mathcal {O}(k \log (k+d) )} \cdot n^{\mathcal {O}(1)}$$ , and remains NP-hard for every constant $$d \ge 2$$ (Acta Informatica (2014)). We present a different FPT algorithm that runs in time $$2^{\mathcal {O}(dk)} \cdot n^{\mathcal {O}(1)}$$ . In particular, our algorithm runs in time $$2^{\mathcal {O}(k)} \cdot n^{\mathcal {O}(1)}$$ , for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by $$k + d$$ . We answer this question in the negative and prove that it does not admit a polynomial compression unless $$\textsf {NP}\subseteq \textsf {coNP}/poly$$ .

Graph Contraction on Attribute-Based Coloring

Graph structures are nowadays pervasive in Big Data. It is often useful to regroup such data in clusters, according to distinctive node features, and use a representative element for each cluster. In many real-world cases, clusters can be identified by a set of connected vertices that share the result of some categorical function, i.e. a mapping of the vertices into some categorical representation that takes values in a finite set C. As an example, we can identify contiguous terrains with the same discrete property on a geographical map, leveraging Space Syntax. In this case, thematic areas within cities are labelled with different colors and color zones are analysed by means of their structure and their mutual interactions. Contracted graphs can help identify issues and characteristics of the original structures that were not visible before.This paper introduces and discusses the problem of contracting possibly large colored graphs into much smaller representatives. It also provides a novel serial but parallelizable algorithm to tackle this task. Some initial performance plots are given and discussed together with hints for future development.

Some Results on Colored Network Contraction

Networks are pervasive in computer science and in real world applications. It is often useful to leverage distinctive node features to regroup such data in clusters, by making use of a single representative node per cluster. Such contracted graphs can help identify features of the original networks that were not visible before. As an example, we can identify contiguous nodes having the same discrete property in a social network. Contracting a graph allows a more scalable analysis of the interactions and structure of the network nodes. This paper delves into the problem of contracting possibly large colored networks into smaller and more easily manageable representatives. It also describes a simple but effective algorithm to perform this task. Extended performance plots are given for a range of graphs and results are detailed and discussed with the aim of providing useful use cases and application scenarios for the approach

A Simple Certifying Algorithm for 3-Edge-Connectivity

A linear-time certifying algorithm for 3-edge-connectivity is presented. Given an undirected graph G, if G is 3-edge-connected, the algorithm generates a construction sequence as a positive certificate for G. Otherwise, the algorithm decomposes G into its 3-edge-connected components and at the same time generates a construction sequence for each connected component as well as the bridges and a cactus representation of the cut-pairs in G. All of these are done by making only one pass over G using an innovative graph contraction technique. Moreover, the graph need not be 2-edge-connected.

Deletion and contraction in configuration spaces of graphs

The aim of this article is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and identify the homotopy cofibers in terms of configuration spaces of simpler graphs. The construction's main benefit lies in making the operations functorial - in particular, graph minors give rise to compatible maps at the level of fundamental groups as well as generalized (co)homology theories. As applications we provide a long exact sequence for half-edge deletion in any generalized cohomology theory, compatible with cohomology operations such as the Steenrod and Adams operations, allowing for inductive calculations in this general context. We also show that the generalized homology of unordered configuration spaces is finitely generated as a representation of the opposite graph minor category.

K(Na,K)Na2[Cu2(SO4)4]: a new highly porous anhydrous sulfate and evaluation of possible ion migration pathways

Alkali copper sulfates form a rapidly developing family of inorganics. Herein, we report synthesis and crystal structure, and evaluate possible ion migration pathways for a novel Na-K-Cu anhydrous sulfate, K(Na,K)Na2[Cu2(SO4)4]. The CuO7 and SO4 polyhedra share common vertices and edges to form [Cu2(SO4)4]4− wide ribbons, which link to each other via common oxygen atoms forming the host part of the structure. Four guest alkali sites are occupied by solely K+, mixture of K+ and Na+, and solely Na+, which agrees well with the size of the cavities. The crystal structure of K(Na,K)Na2[Cu2(SO4)4] contains two symmetry-independent Cu sites with [4+1+(2)] coordination environments. The overall coordination polyhedra of Cu2+ can be considered as `octahedra with one split vertex'. A similar coordination mode was observed also in some other multinary copper sulfates, mostly of the mineral world. These coordination modes were reviewed and five types of CuO7 polyhedra are identified. CuO7 polyhedra are almost restricted to copper sulfates and phosphates. It was found that a larger amount of the smaller SO4 2− and PO4 3− anions can cluster around a single Cu2+ cation; in addition, for such relatively small anions, both mono (κ1) and bidentate (κ2) coordination modes to the Cu2+ are possible. The correlation between crystallographic characteristics and bond valence energies showed that the new copper sulfate framework, [Cu2(SO4)4]4−, contains one interconnected path suitable for Na+ mobility at tolerable activation energies and that K(Na,K)Na2[Cu2(SO4)4] can be considered as a potential candidate for novel Na-ion conductors.

Proof of a conjecture on Wiener index and eccentricity of a graph due to edge contraction

For a connected graph G, the Wiener index, denoted by W(G), is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by ɛ(G), is the sum of the eccentricity of individual vertices. In Darabi et al. (2021), the authors posed a conjecture which states that given a graph G with at least three vertices, the difference between W(G) and ɛ(G) decreases when an edge is contracted and proved that the conjecture is true when e is a bridge. In this manuscript, we confirm that the conjecture is true for any connected graph G with at least three vertices irrespective of the nature of the edge chosen.

A Study on Hybrid Hierarchical Network Representation Learning

Network representation learning (NRL) aims to convert nodes of a network into vector forms in Euclidean space. The information of a network is needed to be preserved as much as possible when NRL converts nodes into vector representation. A hybrid approach proposed in this paper is a framework to improve other NRL methods by considering the structure of densely connected nodes (community-like structure). HARP [1] is to contract a network into a series of contracted networks and embed them from the high-level contracted network to the low-level one. The vector representation (or embedding) for a high-level contracted network is used to initialize the learning process of a low-level contracted graph hierarchically. In this method (Hybrid Approach), HARP is revised by using a well-designed initialization process on the most high-level contracted network to preserve more community-like structure information.

A Digital Twin-Oriented Lightweight Approach for 3D Assemblies

In the design and operation scenarios driven by Digital Twins, large computer-aided design (CAD) models of production line equipment can limit the real-time performance and fidelity of the interaction between digital and physical entities. Digital CAD models often consist of combined parts with characteristics of discrete folded corner planes. CAD models simplified to a lower resolution by current mainstream mesh simplification algorithms might suffer from significant feature loss and mesh breakage, and the interfaces between the different parts cannot be well identified and simplified. A lightweight approach for common CAD assembly models of Digital Twins is proposed. Based on quadric error metrics, constraints of discrete folded corner plane characteristics of Digital Twin CAD models are added. The triangular regularity in the neighborhood of the contraction target vertices is used as the penalty function, and edge contraction is performed based on the cost. Finally, a segmentation algorithm is employed to identify and remove the interfaces between the two CAD assembly models. The proposed approach is verified through common stereoscopic warehouse, robot base, and shelf models. In addition, a scenario of a smart phone production line is applied. The experimental results indicate that the geometric error of the simplified mesh is reduced, the frame rate is improved, and the integrity of the geometric features and triangular facets is effectively preserved.

β-Hydroxybutyrate impairs neutrophil migration distance through activation of a protein kinase C and myosin light chain 2 signaling pathway in ketotic cows

Ketosis in dairy cows often occurs in the peripartal period and is accompanied by immune dysfunction. High concentrations of β-hydroxybutyrate (BHB) in peripheral blood during ketosis are closely related to the impairment of polymorphonuclear neutrophil (PMN) chemotaxis and contribute to immune dysfunction. The specific effect of BHB on PMN chemotaxis in dairy cows and the underlying molecular mechanisms are unclear. Here, 30 multiparous cows (within 3 wk postpartum) classified based on serum BHB as control (n = 15, BHB <0.6 mM) or clinically ketotic (n = 15, BHB >3.0 mM) were used. Blood samples were collected before feeding, and the isolated PMN were treated with platelet-activating factor for 0.5 h to activate their migration. Scanning electron microscopy revealed a longer tail in PMN of ketotic cows. In addition, the phosphorylation and transcription levels of myosin light chain 2 (MLC2) increased in PMN of ketotic cows. Polymorphonuclear neutrophils from control dairy cows were incubated with 3.0 mM BHB for different times in vitro, and 6 h was selected as the proper duration of BHB stimulation according to its inhibition effect on PMN migration using an under-agarose PMN chemotaxis model. Similarly, BHB stimulation in vitro resulted in inhibition of migration distance and deviation of migration direction of PMN, as well as a longer tail in morphology in the scanning electron microscope data, suggesting that BHB-induced PMN migration inhibition may be mediated by impairing the trailing edge contraction. To confirm this hypothesis, sotrastaurin (Sotra)-a specific inhibitor of protein kinase C (PKC), which is the core regulator of cell contraction-was used with or without BHB treatment in vitro. Sotra was pretreated 0.5 h before BHB treatment. Accordingly, BHB treatment increased the phosphorylation level of PKC and MLC2, the protein abundance of RhoA and rho-kinase 1 (ROCK1), and the mRNA abundance of PRKCA, MYL2, RHOA, and ROCK1 in PMN. In contrast, these effects of BHB on PMN were dampened by Sotra. As demonstrated by immunofluorescence experiments in vitro, the BHB-induced inhibition of trailing edge contraction of PMN was relieved by Sotra. In addition, Sotra also dampened the effects of BHB on PMN migration in vitro. Furthermore, as verified by in vivo experiments, compared with the control cows, both abundance and activation of PKC signaling were enhanced in PMN of ketotic cows. Overall, the present study revealed that high concentrations of blood BHB impaired PMN migration distance through inhibition of the trailing edge contraction, mediated by enhancing the activation of PKC-MLC2 signaling. These findings help explain the dysfunctional immune state in ketotic cows and provide information on the pathogenesis of infectious diseases secondary to ketosis.

Reducing the domination number of [formula omitted]-free graphs via one edge contraction

In this note, we consider the following problem: given a connected graph G, can we reduce the domination number of G by using only one edge contraction? We show that the problem is polynomial-time solvable on (P3+kP2)-free graphs for any k≥0 which can be combined with former results to obtain a complexity dichotomy of the problem on H-free graphs.

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε &gt; 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ &gt; 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

Eliminating isomorphism identification method for synthesizing nonfractionated kinematic chains based on graph similarity

A novel method for eliminating isomorphism identification is proposed to improve synthesis efficiency and synthesize planar nonfractionated kinematic chain (KCs) automatically. This method is based on the vertex insertion of contracted graphs. First, similar edges of contracted graphs are divided into groups, in which the similar edges are found and their characteristic matrices are calculated. The edge types are divided based on whether or not isomerism occurs after vertices are inserted. Then, the vertices are inserted into contracted graphs according to the edge condition. In this process, all isomerism caused by the location and number of inserted vertices is reserved, and the property change of similar edges is checked. Lastly, the rigid subchains of remaining isomerism are distinguished. Contracted graphs with four independent loops and some of their inserted vertices are presented in appendix. A complete set of nonfractionated KCs with up to seven independent loops and three degrees of freedom is also provided. The veracity and efficiency of the method are confirmed by conducting a comparative analysis between this synthesis and other literature results.

In vitro and in vivo Effect of Platelet-Rich Plasma-Based Autologous Topical Serum on Cutaneous Wound Healing

Introduction: Skin injury and wound healing is an inevitable event during lifetime. However, several complications may hamper the regeneration of the cutaneous tissue and lead to a chronic profile that prolongs patient recovery. Platelet-rich plasma is rising as an effective and safe alternative to the management of wounds. However, this technology presents some limitations such as the need for repeated blood extractions and health-care interventions. Objective: The aim of this study was to assess the use of an endogenous and storable topical serum (ES) derived from plasma rich in growth factors promoting wound healing, and to obtain preliminary data regarding its clinical and experimental effect over ulcerated skin models and patient care. Methods: Human dermal fibroblast and 3D organotypic ulcerated skin models were used to assess ES over the main mechanisms of wound healing including cell migration, edge contraction, collagen synthesis, tissue damage, extracellular matrix remodeling, cell death, metabolic activity, and histomorphometry analysis. Additionally, 4 patients suffering from skin wounds were treated and clinically assessed. Results: ES promoted dermal fibroblast migration, wound edge contraction, and collagen synthesis. When topically applied, ES increased collagen and elastin deposition and reduced tissue damage. The interstitial edema, structural integrity, and cell activity were also maintained, and apoptotic levels were reduced. Patients suffering from hard-to-heal wounds of different etiologies were treated with ES, and the ulcers healed completely within few weeks with no reported adverse events. Conclusion: This preliminary study suggests that ES might promote cutaneous wound healing and may be useful for accelerating the re-epithelization of skin ulcers.

MFIF-GAN: A new generative adversarial network for multi-focus image fusion

Multi-Focus Image Fusion (MFIF) is a promising image enhancement technique to generate all-in-focus images meeting visual needs, and it is a precondition for other computer vision tasks. One emergent research trend in MFIF involves approaches to avoiding a defocus spread effect (DSE) around a focus/defocus boundary (FDB). This study proposes a generative adversarial network for MFIF tasks called MFIF-GAN, to attenuate the DSE by generating focus maps in which the foreground region is correctly larger than corresponding objects. A Squeeze and Excitation residual module is employed in the proposed network. By combining prior knowledge of a training condition, the network is trained on a synthetic dataset based on an α-matte model. In addition, reconstruction and gradient regularization terms are combined in the loss functions to enhance boundary details and improve the quality of fused images. Extensive experiments demonstrate that the MFIF-GAN outperforms eight state-of-the-art (SOTA) methods in visual perception and quantitative analysis, as well as efficiency. Moreover, an edge diffusion and contraction module is proposed to verify that focus maps generated by our method are accurate at the pixel level.

RETRACTED ARTICLE: A New Derivation Method of Contracted Graphs with Pentagonal Links Plus Other Links for Type Synthesis of Closed Mechanisms

In this paper, a derivation method is presented for the valid contracted graphs (CGs) with pentagonal links plus quaternary links Q and ternary links T using starting vertax arrays strings (SVASs) for type synthesis of parallel mechanisms. First, some concepts on the CGs, starting vertax array strings (SVASs), the basic contracted graphs (BCGs) are explained. Second, a method is presented for deriving the valid CGs according to the BCGs of the valid SVASs. According to the connections of starting vertax, the BCGs of the valid SVASs are obtained. Then, according to the principle of permutation and combination, all the valid CGs are derived in a basic link queue. Third, the synthesis of CGs with pentagonal links P, quaternary links Q, and ternary links T for closed mechanisms is studied, and many valid CGs with P, Q and T are derived by the BCGs of the SVASs.

Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.