Edge Cut Research Articles

Edge cut splitting formulas for Tutte–Grothendieck invariants

Tutte–Grothendieck invariants of graphs are mappings from a class of graphs to a commutative ring that are characterized recursively by contraction-deletion rules. Well known examples are the Tutte, chromatic, tension and flow polynomials. Suppose that an edge cut C divides a graph G into two parts G1, G1′ and that G1, G1′ are the sets of minors of G whose edge sets consist of C and edges of G1, G1′, respectively. We study determinant formulas evaluating a Tutte–Grothendieck invariant of G from the Tutte–Grothendieck invariants of graphs from G1 and G1′.

Solving Graph Laplacian Systems Through Recursive Partitioning and Two-Grid Preconditioning

We present a parallelizable direct method for computing the solution to graph Laplacian-based linear systems derived from graphs that can be hierarchically bipartitioned with small edge cuts. For a graph of size $n$ that can be recursively partitioned into equal-sized sets with constant-size edge cuts, our method decomposes a graph Laplacian in time $O(n \log n)$ and then uses that decomposition to perform a linear solve in time $O(n \log n)$. Many real-world graphs, however, do not possess this property. So, we use this technique to design a preconditioner for graph Laplacians that do not have this property. Finally, we augment this preconditioner with a two-grid method that accounts for much of the preconditioner's weaknesses. We present an analysis of this method, as well as a general theorem for the condition number of a general class of two-grid support graph-based preconditioners. Numerical experiments illustrate the performance of the studied methods.

COMPARISON OF THE STRENGTH OF NORMAL AND EDGE-CUT TENSILE SPECIMENS OF STYRENE–BUTADIENE RUBBER AND NATURAL RUBBER WITH SIMILAR CROSSLINK DENSITY

ABSTRACT Edge cuts of various depths, c, were introduced into tensile specimens of similarly crosslinked (ρc ≈ 6 (10−5) moles of crosslinks/mL) gum (unfilled) natural rubber (NR) and gum styrene–butadiene rubber (SBR). When specimens contained no intentional cut (c = 0, i.e., normal tensile strength), the NR is about 10 times as strong as the SBR. This difference is due to the stereoregularity (cis-1,4-polyisoprene) of NR, enabling it to strain crystallize when deformed. On the other hand, SBR has an irregular, amorphous microstructure that renders it incapable of crystallization. However, for test pieces that contain an edge cut, the strength, σbc, of the NR relative to that of the SBR depends strongly on cut depth. When c ≈ 0.2 mm, the strength of both vulcanizates is reduced about 50%, and hence the ratio of strengths remains about an order of magnitude. But, with a further increase in c, the SBR exhibits a steady decrease in σbc, while the strength of the NR drops discontinuously by nearly a factor of four when c ≈ 1.7 mm. Now, the NR is only about two and one half times as strong as the SBR. Extrapolation of σbc to larger c results in a predicted cut depth c ≈ 4.3 mm, at which the strength of the NR and SBR would be similar. At sufficiently large cut depth, it appears that strain rate at the cut tip is high enough and breaking strain low enough that rupture occurs before significant strain crystallization commences.

Steady state shift damage localization

The accuracy of the identified modal model is the weak link in many vibration based damage localization schemes. This paper presents a localization approach that avoids identification by operating with two frequency domain subspaces, one obtained by Fourier transformation of output measurements and the other from a model of the reference state and a postulated damage distribution. The approach differs from a model updating framework in that only the damage distribution, and not the extent, enters the formulation. The method operates on the premise that the loads are time limited, have an invariant distribution in space and requires that the histories be repeatable, or that the excitation be a single history, in which case repeatability is not necessary. The time histories of the excitation are not used in the technique and therefore need not be known. It is shown that multiple damages can be considered without combinatorics if $$e \ge \kappa$$ where e is the number of available actuators and $$\kappa$$ is the rank of the change in the transfer matrix due to damage. The constraint on the number of measurements, m, is $$m \ge \kappa .$$ The method, designated as the steady state shift damage localization (S3DL) is experimentally tested on an aluminum plate where damage is simulated by mass additions and by an edge cut.

Edge Cut Process for Reducing Ni Content at Channel Edge Region in Metal Induced Lateral Crystallization Poly-Si TFTs

Nickel silicide is main issue in Polycrystalline silicon Thin Film Transistor (TFT) which is made by Metal Induced Lateral Crystallization (MILC) method. This Nickel silicide acts as a defect center, and this defect is one of the biggest reason of the high leakage current. In this research, we fabricated polycrystalline TFTs with novel method called Edge Cut (EC). With this new fabrication method, we assumed that nickel silicide at the edge of the channel region is reduced. Electrical properties are measured and trap state density also calculated using Levinson & Proano method.

The Minset-Poset Approach to Representations of Graph Connectivity

Various instances of the minimal-set poset (minset-poset for short) have been proposed in the literature, e.g., the representation of Picard and Queyranne for all st -minimum cuts of a flow network. We begin with an explanation of why this poset structure is common. We show any family of sets F that can be defined by a “labelling algorithm” (e.g., the Ford-Fulkerson labelling algorithm for maximum network flow) has an algorithm that constructs the minset poset for F . We implement this algorithm to efficiently find the nodes of the poset when F is the family of minimum edge cuts of an unweighted graph; we also give related algorithms to construct the entire poset for weighted graphs. The rest of the article discusses applications to edge- and vertex connectivity, both combinatorial and algorithmic, that we now describe. For digraphs, a natural interpretation of the minset poset represents all minimum edge cuts. In the special case of undirected graphs, the minset poset is proved to be a variant of the well-known cactus representation of all mincuts. We use the poset algorithms to construct the cactus representation for unweighted graphs in time O ( m +λ 2 n log (n/λ)) (λ is the edge connectivity) improving the previous bound O (λ n 2 ) for all but the densest graphs. We also construct the cactus representation for weighted graphs in time O ( nm log( n 2 / m )), the same bound as a previously known algorithm but in linear space O ( m ). The latter bound also holds for constructing the minset poset for any weighted digraph; the former bound also holds for constructing the nodes of that poset for any unweighted digraph. The poset is used in algorithms to increase the edge connectivity of a graph by adding the fewest edges possible. For directed and undirected graphs, weighted and unweighted, we achieve the time of the preceding two bounds, i.e., essentially the best-known bounds to compute the edge connectivity itself. Some constructions of minset posets for graph rigidity are also sketched. For vertex connectivity, the minset poset is proved to be a slight variant of the dominator tree. This leads to an algorithm to construct the dominator tree in time O ( m ) on a RAM. (The algorithm is included in the appendix, since other linear-time algorithms of similar simplicity have recently been presented.)

A framework to compare topology algorithms in multi-channel multi-radio wireless mesh networks

Creating an optimal topology for a multi-channel multi-radio wireless mesh network by an algorithm is a balancing act between different metrics such as bandwidth, delay, and redundancy. We propose a framework to quickly evaluate and compare resulting network graphs of different topology algorithms, both distributed and centralized, in wireless mesh networks. This framework complements to graph analysis and network simulation. The metrics presented in this paper, are both data flow (such as bandwidth capacity, delay, and loss) and structural characteristics (such as minimal edge and node cut) related. Each presented metric can be solved using a linear programming with the weighted incidence matrix of the network graph and the protocol interference model. The framework uses matrix operations, which are well established, making the framework unambiguous and easy to implement. We demonstrate the framework by comparing topology algorithms with increasing complexity and by comparing topology algorithms found in literature.

Moria One-Use Plus sub-Bowman’s keratomileusis head: a useful tool in the refractive surgeon’s armamentarium

Patients and methods A total of 500 eyes of 250 patients underwent laser in situ keratomileusis (LASIK) surgery using the One-Use Plus sub-Bowman’s keratomileusis head for the creation of an ultrathin LASIK flap. During the follow-up at 1, 6, 12, and 24 months, both uncorrected visual acuity (UCVA) and best-corrected visual acuity were recorded. A keratoconus screening corneal topography was carried out during the last follow-up visit. Results A total of 500 eyes of 250 myopic patients were included in this study. There were 115 male patients and 135 female patients. The mean age was 27.5 years (range: 18–44 years). The mean preoperative refractive error was −5.0 diopter sphere (DS) (range: −3 to −9 DS). The mean cylindrical error was −2.5 diopter cylinder (DC) (range: −0.75 to −4.5 DC). The mean UCVA was 0.07 (range: 0.01–0.3), and it improved to 1.0 (range: 0.8–1.25) on day 1 postoperatively. At the end of the follow-up, the mean UCVA was 1.0 and the mean best-corrected visual acuity was 1.0. The mean stromal residual bed thickness was 362 μm (range: 304–466 μm), and the mean central flap thickness was 102 μm (range: 82–120 μm). Complications were perilimbal bleeding in 15 eyes and fine irregular stromal edge cuts at the sides of the hinge in 10 eyes. There were no epithelial defects, no epithelial heaping or sliding, no irregular stromal bed surface cuts, no free caps or button holes, or incomplete flaps. Diffuse lamellar keratitis occurred in one eye of one patient, and no slipped flaps, macrostriae or microstriae, or epithelial downgrowth was seen during the follow-up period. At the end of the follow-up period none of our patients had topographic evidence of keratoconus. Conclusion The One-Use Plus sub-Bowman’s keratomileusis head is a safe and effective method for creating an ultrathin LASIK flap and allows the surgeon to treat higher errors of refraction while maintaining a higher safety margin by leaving more residual stromal tissue.

Strong cliques and equistability of EPT graphs

In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.

Algorithms solving the Matching Cut problem

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete. This paper provides a first branching algorithm solving Matching Cut in time O⁎(2n/2)=O⁎(1.4143n) for an n-vertex input graph, and shows that Matching Cut parameterized by the vertex cover number τ(G) can be solved by a single-exponential algorithm in time 2τ(G)O(n2). Moreover, the paper also gives a polynomially solvable case for Matching Cut which covers previous known results on graphs of maximum degree three, line graphs, and claw-free graphs.

A multilevel tabu search algorithm for balanced partitioning of unstructured grids

SummaryThis paper presents a multilevel algorithm for balanced partitioning of unstructured grids. The grid is partitioned such that the number of interface elements is minimized and each partition contains an equal number of grid elements. The partition refinement of the proposed multilevel algorithm is based on iterative tabu search procedure. In iterative partition refinement algorithms, tie‐breaking in selection of maximum gain vertices affects the performance considerably. A new tie‐breaking strategy in the iterative tabu search algorithm is proposed that leads to improved partitioning quality. Numerical experiments are carried out on various unstructured grids in order to evaluate the performance of the proposed algorithm. The partition results are compared with those produced by the well‐known partitioning package Metis and k‐means clustering algorithm and shown to be superior in terms of edge cut, partition balance, and partition connectivity. Copyright © 2015 John Wiley & Sons, Ltd.

On the structure of vertex cuts separating the ends of a graph

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an infinite graph. In this paper we show that minimal vertex cuts separating the ends of a graph can be encoded by a succulent, a mild generalization of a cactus that is still tree-like. We go on to show that the earlier cactus results follow from our work.

A Structure Theorem for Strong Immersions

AbstractA graph H is strongly immersed in G if H is obtained from G by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge‐disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We describe the structure of graphs avoiding a fixed graph as a strong immersion. The theorem roughly states that a graph which excludes a fixed graph as a strong immersion has a tree‐like decomposition into pieces glued together on small edge cuts such that each piece of the decomposition has a path‐like linear decomposition isolating the high degree vertices.

Sufficient conditions for triangle-free graphs to be super k-restricted edge-connected

An edge cut S of a connected graph G=(V,E) is a k-restricted edge cut if every component of G−S contains at least k vertices. A graph is said to be super k-restricted edge-connected if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. Let k be a positive integer, and let G be a connected triangle-free graph of order n≥2k. In this paper, we prove that if the minimum degree δ(G)≥k+1−(−1)k and there are at least k+1+(−1)k2 common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k-restricted edge-connected.

A new heuristic for detecting non-Hamiltonicity in cubic graphs

We analyse a polyhedron which contains the convex hull of all Hamiltonian cycles of a given undirected connected cubic graph. Our constructed polyhedron is defined by polynomially-many linear constraints in polynomially-many continuous (relaxed) variables. Clearly, the emptiness of the constructed polyhedron implies that the graph is non-Hamiltonian. However, whenever a constructed polyhedron is non-empty, the result is inconclusive. Hence, the following natural question arises: if we assume that a non-empty polyhedron implies Hamiltonicity, how frequently is this diagnosis incorrect? We prove that, in the case of bridge graphs, the constructed polyhedron is always empty. We also demonstrate that some non-bridge non-Hamiltonian cubic graphs induce empty polyhedra as well. We compare our approach to the famous Dantzig–Fulkerson–Johnson relaxation of a TSP, and give empirical evidence which suggests that the latter is infeasible if and only if our constructed polyhedron is also empty. By considering special edge cut sets which are present in most cubic graphs, we describe a heuristic approach, built on our constructed polyhedron, for which incorrect diagnoses of non-Hamiltonian graphs as Hamiltonian appear to be very rare. In particular, for cubic graphs containing up to 18 vertices, only four out of 45,982 undirected connected cubic graphs were so misdiagnosed. By constrast, we demonstrate that an equivalent heuristic, when built on the Dantzig–Fulkerson–Johnson relaxation of a TSP, is mostly unsuccessful in identifying additional non-Hamiltonian graphs. These empirical results suggest that polynomial algorithms based on our constructed polyhedron may be able to correctly identify Hamiltonicity of a cubic graph in all but rare cases.

Extremal aspects of the Erdős–Gallai–Tuza conjecture

Erdős, Gallai, and Tuza posed the following problem: given an n-vertex graph G, let τ1(G) denote the smallest size of a set of edges whose deletion makes G triangle-free, and let α1(G) denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that α1(G)+τ1(G)≤n2/4? We also consider a variant on this conjecture: if τB(G) is the smallest size of an edge set whose deletion makes G bipartite, does the stronger inequality α1(G)+τB(G)≤n2/4 always hold?By considering the structure of a minimal counterexample to each version of the conjecture, we obtain two main results. Our first result states that any minimum counterexample to the original Erdős–Gallai–Tuza Conjecture has “dense edge cuts”, and in particular has minimum degree greater than n/2. This implies that the conjecture holds for all graphs if and only if it holds for all triangular graphs (graphs where every edge lies in a triangle). Our second result states that α1(G)+τB(G)≤n2/4 whenever G has no induced subgraph isomorphic to K4−, the graph obtained from the complete graph K4 by deleting an edge. Thus, the original conjecture also holds for such graphs.

Extension of the elasticity-conductivity cross-property connections to impedance of plane microcracked structural elements

The paper focuses on empirical extension of the cross-property connection between changes in elasto-static compliance and electrical resistance associated with plane microcracking in structural elements to changes in electrical impedance. Theoretical underlying principles of impedance are discussed and experimental measurements are taken to compare AC impedance, DC resistance and changes in resistivity predicted by the cross-property connection before and after subjecting test specimens to an edge cut. If the reactive components of impedance are sufficiently isolated, experimental results show a strong correlation between changes in real-part resistive impedance, directly extending the cross-property connection to impedance.

Vertex Cuts

AbstractGiven a connected graph, in many cases it is possible to construct a structure tree that provides information about the ends of the graph or its connectivity. For example Stallings' theorem on the structure of groups with more than one end can be proved by analyzing the action of the group on a structure tree and Tutte used a structure tree to investigate finite 2‐connected graphs, that are not 3‐connected. Most of these structure tree theories have been based on edge cuts, which are components of the graph obtained by removing finitely many edges. A new axiomatic theory is described here using vertex cuts, components of the graph obtained by removing finitely many vertices. This generalizes Tutte's decomposition of 2‐connected graphs to k‐connected graphs for any k, in finite and infinite graphs. The theory can be applied to nonlocally finite graphs with more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a decomposition for a group acting on such a graph, generalizing Stallings' theorem. Further applications include the classification of distance transitive graphs and k‐CS‐transitive graphs.

Degree conditions for graphs to be maximally [formula omitted]-restricted edge connected and super [formula omitted]-restricted edge connected

For a connected graph G=(V,E), an edge set S⊆E(G) is called a k-restricted edge cut of G if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex subsets X,Y of G, define [X,Y]={xy∈E(G):x∈X,y∈Y} and define ξk(G)=min{|[X,X¯]|:X⊆V(G),|X|=k,G[X]is connected}, where X¯=V(G)∖X. G is λk-optimal if λk(G)=ξk(G). Furthermore, G is super-λk if every minimum k-restricted edge cut of G isolates a connected subgraph with order k. The k-restricted edge connectivity is an important index to estimate the reliability of networks. In this paper, some degree conditions for graphs to be maximally k-restricted edge connected and super k-restricted edge connected are given.

Formula omitted]-restricted edge connectivity in [formula omitted]-clique-free graphs

Let G be a graph with vertex set V(G) and edge set E(G). An edge subset S⊆E(G) is called a k-restricted edge cut if G−S is not connected and every component of G−S has at least k vertices. The k-restricted edge connectivity of a connected graph G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let [X,X̄] denote the set of edges between a vertex set X⊂V(G) and its complement X̄=V(G)∖X. A vertex set X⊂V(G) is called a λk-fragment if [X,X̄] is a minimum k-restricted edge cut of G. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X]is connected}. In this work, we give a lower bound on the cardinality of λk-fragments of a graph G satisfying λk(G)<ξk(G) and containing no (p+1)-cliques. As a consequence of this result, we show a sufficient condition for a graph G with λk(G)=ξk(G).