Edge Cut Research Articles

Graphical Metrics for Analyzing District Maps

For the past several decades, political scientists have sought to understand the impact of legislative redistricting and gerrymandering on a variety of outcomes. However, traditional metrics such as compactness scores and newer metrics such as aggregated simulations impose very strong assumptions that make their use difficult. In this study, we propose a new graphical framework for analyzing districts that relaxes current assumptions while allowing analysts to focus on the choices that redistricting parties may potentially make. We then leverage the newest advances in district simulation algorithms to extend this framework to propose four new metrics. These new metrics are Edge Cut Growth (ECG), Excess Edge (EE), and Edge per District Gain (EDG), and Internal Boundary Growth (IBG). These new metrics are then compared to several existing metrics, allowing us to test the attributes that our approach is similar to. In doing so, we demonstrate that the four new metrics are best seen as theoretical and technical advances on current metrics that focus on district geometry.

Monochromatic disconnection of graphs

For an edge-colored graph G, we call an edge–cut M of G monochromatic if the edges of M are colored with the same color. The graph G is called monochromatically disconnected if any two distinct vertices of G are separated by a monochromatic edge–cut. For a connected graph G, the monochromatic disconnection number of G, denoted by md(G), is the maximum number of colors that are needed in order to make G monochromatically disconnected. We show that almost all graphs have monochromatic disconnection numbers equal to 1. We also obtain the Nordhaus–Gaddum-type results for md(G).

Reliability Analysis of Alternating Group Graphs and Split-Stars

AbstractGiven a connected graph $G$ and a positive integer $\ell $, the $\ell $-extra (resp. $\ell $-component) edge connectivity of $G$, denoted by $\lambda ^{(\ell )}(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of edges whose removal from $G$ results in a disconnected graph so that every component has more than $\ell $ vertices (resp. so that it contains at least $\ell $ components). This naturally generalizes the classical edge connectivity of graphs defined in term of the minimum edge cut. In this paper, we proposed a general approach to derive component (resp. extra) edge connectivity for a connected graph $G$. For a connected graph $G$, let $S$ be a vertex subset of $G$ for $G\in \{\Gamma _{n}(\Delta ),AG_n,S_n^2\}$ such that $|S|=s\leq |V(G)|/2$, $G[S]$ is connected and $|E(S,G-S)|=\min \limits _{U\subseteq V(G)}\{|E(U, G-U)|: |U|=s, G[U]\ \textrm{is connected}\ \}$, then we prove that $\lambda ^{(s-1)}(G)=|E(S,G-S)|$ and $\lambda _{s+1}(G)=|E(S,G-S)|+|E(G[S])|$ for $s=3,4,5$. By exploring the reliability analysis of $AG_n$ and $S_n^2$ based on extra (component) edge faults, we obtain the following results: (i) $\lambda _3(AG_n)-1=\lambda ^{(1)}(AG_n)=4n-10$, $\lambda _4(AG_n)-3=\lambda ^{(2)}(AG_n)=6n-18$ and $\lambda _5(AG_n)-4=\lambda ^{(3)}(AG_n)=8n-24$; (ii) $\lambda _3(S_n^2)-1=\lambda ^{(1)}(S_n^2)=4n-8$, $\lambda _4(S_n^2)-3=\lambda ^{(2)}(S_n^2)=6n-15$ and $\lambda _5(S_n^2)-4=\lambda ^{(3)}(S_n^2)=8n-20$. This general approach maybe applied to many diverse networks.

Evolutionary multi-level acyclic graph partitioning

Directed graphs are widely used to model data flow and execution dependencies in streaming applications. This enables the utilization of graph partitioning algorithms for the problem of parallelizing execution on multiprocessor architectures under hardware resource constraints. However due to program memory restrictions in embedded multiprocessor systems, applications need to be divided into parts without cyclic dependencies. We found that this can be done by a subsequent second graph partitioning step with an additional acyclicity constraint. We have four main contributions. First, we show that this more constrained version of the graph partitioning problem is NP-complete and present linear time heuristics. We then integrate them into an existing multi-level graph partitioning framework to better handle large graphs. This achieves a 9% reduction of the edge cut compared to the previous single-level algorithm. Based on this, we engineer an evolutionary algorithm to further reduce the cut, achieving a 30% reduction on average compared to the state of the art. Finally, we integrate the partitioning heuristics into a graph compiler for an embedded multiprocessor architecture and show that this can reduce the amount of communication for a real-world imaging application and thereby accelerate it by an average of 11%. It is shown that the compiler can emit optimized code for vastly different hardware platforms using the heuristics. In addition, we demonstrate how a custom fitness function for the evolutionary algorithm can be used to optimize other objectives like load balancing if the communication volume is not predominantly important on a given hardware platform.

The component (edge) connectivity of shuffle-cubes

Component (edge) connectivity is a generalization of traditional (edge) connectivity. Let F be a vertex set (resp., an edge set), if G−F has at least g components, F is a g-component (edge) cut. The g-component (edge) connectivity of graph G is the minimum size of the g-component (edge) cut. In this paper, we study the g-component (edge) connectivity of shuffle-cubes for small g.

Mitigating Wafer Edge Cut Defect through Tensionless Tape Lamination Technique

Thinner and smaller packages require thinner vertical structure of the integrated circuit (IC) design with the wafer playing essential role in package thinning. As the wafer goes thinner, problems may occur in the pre-assembly or wafer preparation. With the introduction of new pre-assembly technology such as laser die attach film (DAF) cut and dicing before grinding, technical challenges were expected. The paper focused on eliminating the edge cutting issue by considering the appropriate taping lamination technique. Tensionless lamination helped eliminate the horizontal pressure applied into the tape thus mitigating the edge cutting problem. For future works, the configuration shared in this paper could be applied on wafers with comparable technology.

Fuzzy edge connectivity and fuzzy local edge connectivity with applications to communication networks

The authors (Mathew and Sunitha (2010) [9]) introduced the concept of fuzzy arc cuts for improving the definition of the so-called cut-sets proposed by Yeh and Bang, both generalizing edge cuts from classic graph theory to the fuzzy settings. They argued that non-strong edges need not be considered in a fuzzy arc cut as such edges would not change the strength of connectedness between any vertices. However, we give a counterexample in the paper, which illustrates that this viewpoint may contain some flaws. Based on this observation, a more general definition of fuzzy arc cuts (called fuzzy edge cuts) is given. Moreover, we define fuzzy edge connectivity with two parameters, this definition allows us to talk about edge connectivity more precisely. The concepts of fuzzy local edge cuts and fuzzy local edge connectivity are also introduced. We investigate the relations between fuzzy edge cuts and fuzzy local edge cuts. A characterization for the weight of a minimal fuzzy local edge cut is obtained. Finally, we provide some algorithms to compute fuzzy edge connectivity and fuzzy local edge connectivity of special graphs. Besides, applications of two connectivity parameters in communication networks are presented.

Sufficient conditions for optimally and super m-restricted edge-connected graphs with given girth

ABSTRACT A -connected graph G is optimally m-restricted edge-connected if its m-restricted edge-connectivity equals the minimum m-edge-degree , and super m-restricted edge-connected if all minimum m-restricted edge cuts are trivial. In this study, a bound for the size of graphs with girth greater than any given integer l is obtained firstly. Then the second minimum degree conditions for optimally and super m-restricted edge-connected graphs with girth greater than l are provided. Furthermore, the relationship between optimally m-restricted edge connectivity and optimally -restricted edge connectivity is characterized.

Wafer Preparation Parameter Optimization for Wafer Defects Elimination

In every new technology developed and introduced to the manufacturing floor, particularly in the wafer preparation, entails problems that later induce defects affecting the wafer yield. This paper discusses the optimization of wafer preparation parameters, particularly the tensionless backgrinding tape lamination and DAF cut vacuum control, that mitigates wafer yield detractors such as edge cut, kerf shift and dice pop-out. Based on the evaluation results, tensionless backgrinding lamination affects the kerf shifting and edge cutting, and with proper vacuum control to attain zero dice pop-out process.

Minimum shared‐power edge cut

AbstractWe introduce a problem called minimum shared‐power edge cut (MSPEC). The input to the problem is an undirected edge‐weighted graph with distinguished vertices s and t, and the goal is to find an s‐t cut by assigning “powers” at the vertices and removing an edge if the sum of the powers at its endpoints is at least its weight. The objective is to minimize the sum of the assigned powers. MSPEC is a graph generalization of a barrier coverage problem in a wireless sensor network: given a set of unit disks with centers in a rectangle, what is the minimum total amount by which we must shrink the disks to permit an intruder to cross the rectangle undetected, that is, without entering any disk. This is a more sophisticated measure of barrier coverage than the minimum number of disks whose removal breaks the barrier. We develop a fully polynomial time approximation scheme for MSPEC. We give polynomial time algorithms for the special cases where the edge weights are uniform, or the power values are restricted to a bounded set. Although MSPEC is related to network flow and matching problems, its computational complexity (in P or NP‐hard) remains open.

Some lower bounds for the energy of graphs

The singular values of a matrix A are defined as the square roots of the eigenvalues of A⁎A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A=(BDD⁎C), then E(A)≥2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H)≤E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E(G)≥4m+n(n−2)|det⁡(A)|2n. Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E(G) given in Oboudi (2019) [14].

Artificial Intelligence Knowledge Graph for Dynamic Networks: An Incremental Partition Algorithm

The quick and intelligent requests and answers in artificial intelligence (AI) are inseparable from intelligent data. Knowledge graph makes data more intelligent by establishing association among data, which provides convenience for intelligent search, reasoning and analysis of data. Resource Description Framework (RDF) is an effective data representation model of knowledge graph. This paper takes RDF as the research object and proposes an incremental partition method of intelligent data (IPID) to realize the distributed storage of large-scale AI data. First, IPID gives a mixed object function integrating edge cut and load balancing. Second, IPID devises the initial and incremental partitioning algorithms of RDF. The initial partition divides the original RDF graph into kernel vertices, boundary vertices and free vertices. The boundary and freedom nodes select the kernel vertex with the maximum gain of object function to form a sub-partition. And the incremental partition is in charge of the selection of sub-partition of new and deleted data by the object function. Meanwhile, the incremental partition algorithm would also execute a dynamic adjustment strategy at a certain time interval according to the balance and tightness of sub-partition to satisfy the partitioning object. Finally, IPID is tested on the knowledge graph datasets. The experimental results show that the object function guarantees the quality of knowledge graph partition in edge cut and load balancing, and effectively realizes the incremental partition.

Label Propagation-Based Parallel Graph Partitioning for Large-Scale Graph Data

The increasing importance of graph data in various fields requires large-scale graph data to be processed efficiently. Furthermore, well-balanced graph partitioning is a vital component of parallel/distributed graph processing. The goal of graph partitioning is to obtain a well-balanced graph topology, where the size of each partition is balanced while the number of edge cuts is reduced. Moreover, a graph-partitioning algorithm should achieve high performance and scalability. In this study, we present a novel graph-partitioning algorithm that ensures a high edge cutting quality and excellent parallel processing performance. We apply formulas based on the label propagation algorithm to improve the quality of edge cuts and achieve fast convergence. In our approach, the necessity of applying the label propagation process for all vertices is removed, and the process is applied only for candidate vertices based on a score metric. Our proposed algorithm introduces a stabilization phase in which remote and highly connected vertices are relocated to prevent the algorithm from becoming trapped in local optima. Comparison results show that a prototype based on the proposed algorithm outperforms well-known parallel graph-partitioning frameworks in terms of speed and balance.

Specimen for the Determination of Fatigue Thresholds Under Cyclic Transverse Shear

We develop a basic element of the procedure of construction of the near-threshold part of the diagram of fatigue fracture for metallic materials under the conditions of transverse shear. It is proposed to use a square sample with edge cut and a force scheme of its cyclic loading. We determine the strain-stress state of a square plate with edge crack (regarded as a prototype of the proposed specimen) under the conditions of transverse shear by two independent methods (finite-element method and the method of singular integral equations). The stress intensity factor KII is determined for a broad interval of relative crack lengths with regard for the smooth contact of the crack faces and the presence of friction between the faces.

Investigation into residual stresses inherited during manufacturing of carbon-epoxy composites with delamination

Last decades, an increased attention has been paid to the deep understanding of the processes caused by residual stresses in laminated composites, specifically, shape distortion and possible damage of the product at the stage of manufacturing. The presence of residual stresses in the composite material affects the occurrence of such defects as delamination, warping, as well as the appearance of microcracks in the matrix of the material, which has a significant impact on the static and fatigue strength of the final product made of composite material.In this paper, we consider the change of the distribution of the residual stress field caused by manufacturing process in the specimen with initial delamination defect (crack) due to appearance of free edge cut. Different length of initial crack has been analyzed with well-known composite material AS4/8552-1.The curing model is numerically implemented using ABAQUS finite element modeling package with special user subroutines. The coupled thermal and stress problem under plane strain conditions is solved for the modelling of curing. The history of energy release rates under mode I, II (GI, GII) were analyzed over time during process of polymerization and free edge cut appearance using seam crack technique.A slight increase in the GI and GII values at the crack tip during polymerization and a multiple excess of these values after a mechanical cut is shown. On the contrary obtained by modelling levels of stress components are essential and cannot be ignored in consequent structural analysis. The results of the study can be applied for prediction of residual stresses in composite structure by means of simulation and further understanding the nature of fracture of composites.

Odd Multiway Cut in Directed Acyclic Graphs

We investigate the odd multiway node (edge) cut problem where the input is a graph with a specified collection of terminal nodes, and the goal is to find a smallest subset of non-terminal nodes (edges) to delete so that the terminal nodes do not have an odd length path between them. In an earlier work, Lokshtanov and Ramanujan showed that both odd multiway node cut and odd multiway edge cut are fixed-parameter tractable (FPT) when parameterized by the size of the solution in undirected graphs. In this work, we focus on directed acyclic graphs (DAGs) and design a fixed-parameter algorithm. Our main contribution is a broadening of the shadow-removal framework to address parity problems in DAGs. We complement our FPT results with tight approximability as well as polyhedral results for two terminals in DAGs. Additionally, we show inapproximability results for odd multiway edge cut in undirected graphs even for two terminals.

Matching cut: Kernelization, single-exponential time FPT, and exact exponential algorithms

In a graph, a matching cut is an edge cut that is a matching. Matching Cut, which is known to be NP-complete, is the problem of deciding whether or not a given graph G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O∗(2dc(G))-time and an O∗(2dc¯(G))-time FPT algorithm for Matching Cut, where dc(G) and dc¯(G) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O∗(1.4143n) to O∗(1.3071n) where n is the number of vertices in G. Moreover, we point out that, unless NP⊆coNP∕poly, Matching Cut does not admit a polynomial kernel when parameterized simultaneously by treewidth, the number of edges crossing the cut, and the maximum degree of G.

Phase transitions in normalized cut of social networks

One of the main goals of network analysis and data science is to identify community structures in a network. Community structure is a typical feature of social networks and community detection is considered to be crucial to understand the structure and function of social networks in cyberspace. The community detection problem of social networks can be viewed as a graph partitioning problem whose objective is to identify the external edges as the edge cut that separates the network as two subnetworks by recursive method. The degree-normalized adjacency matrix is more convenient than traditional Laplacian matrix in calculating eigenvalues and eigenvectors. In this work, based on random model we first prove the existence of an abrupt phase transition for community detection in terms of the degree-normalized adjacency matrix with accurate critical value. In detail, the detectability of community detection has changed from high to low when the critical value of phase transition q approaches p1p2 almost surely, where pi(i=1,2) is the intra-community edge connection probability of subnetworks Ci, respectively. Then, the validity of the theoretical results is verified by numerical experiments.

Eulerian Properties of Design Hypergraphs and Hypergraphs with Small Edge Cuts

An Euler tour of a hypergraph is a closed walk that traverses every edge exactly once; if a hypergraph admits such a walk, then it is called eulerian. Although this notion is one of the progenitors of graph theory --- dating back to the eighteenth century --- treatment of this subject has only begun on hypergraphs in the last decade. Other authors have produced results about rank-2 universal cycles and 1-overlap cycles, which are equivalent to our definition of Euler tours. In contrast, an Euler family is a collection of nontrivial closed walks that jointly traverse every edge of the hypergraph exactly once and cannot be concatenated simply. Since an Euler tour is an Euler family comprising a single walk, having an Euler family is a weaker attribute than being eulerian; we call a hypergraph quasi-eulerian if it admits an Euler family. A triple system is a 3-uniform hypergraph in which every pair of vertices lie in exactly $\lambda$ edges together, for some parameter $\lambda$. We first prove that all triple systems with at least two edges are eulerian. We then define a new kind of hypergraph, which we call $\ell$-covering $k$-hypergraphs: these are $k$-uniform hypergraphs in which every $\ell$-subset of the vertices lie in at least one edge together. We are able to show that such hypergraphs are eulerian when $\ell=k-1$ and quasi-eulerian otherwise, as long as they have at least two edges. Finally, we give some constructive results on hypergraphs with small edge cuts. There has been analogous work by other authors on hypergraphs with small vertex cuts. We reduce the problem of finding an Euler tour in a hypergraph to finding an Euler tour in each of the connected components of the edge-deleted subhypergraph, then show how these individual Euler tours can be concatenated.

Coherent network partitions

Graph clustering is widely applied in the analysis of cellular networks reconstructed from large-scale data or obtained from experimental evidence. Here we introduce a new type of graph clustering based on the concept of coherent partition. A coherent partition of a graph G is a partition of the vertices of G that yields only disconnected subgraphs in the complement of G. The coherence number of G is then the size of the smallest edge cut inducing a coherent partition. A coherent partition of G is optimal if the size of the inducing edge cut is the coherence number of G. Given a graph G, we study coherent partitions and the coherence number in connection to (bi)clique partitions and the (bi)clique cover number. We show that the problem of finding the coherence number is NP-hard, but is of polynomial time complexity for trees. We also discuss the relation between coherent partitions and prominent graph clustering quality measures.