Edge Cut Research Articles

Beyond Good Partition Shapes: An Analysis of Diffusive Graph Partitioning

In this paper we study the prevalent problem of graph partitioning by analyzing the diffusion-based partitioning heuristic Bubble-FOS/C, a key component of a practical successful graph partitioner (Meyerhenke et al. in J. Parallel Distrib. Comput. 69(9):750---761, 2009). We begin by studying the disturbed diffusion scheme FOS/C, which computes the similarity measure used in Bubble-FOS/C and is therefore the most crucial component. By relating FOS/C to random walks, we obtain precise characterizations of the behavior of FOS/C on tori and hypercubes. Besides leading to new knowledge on FOS/C (and therefore also on Bubble-FOS/C), these characterizations have been recently used for the analysis of load balancing algorithms (Berenbrink et al. in Proceedings of the 22nd Annual Symposium on Discrete Algorithms, pp. 429---439, 2011). We then regard Bubble-FOS/C, which has been shown in previous experiments to produce solutions with good partition shapes and other favorable properties. In this paper we prove that it computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). This result provides the first substantial theoretical insight why Bubble-FOS/C yields good experimental results in terms of graph partitioning metrics. Moreover, we show that in bisections computed by Bubble-FOS/C, at least one of the two parts is connected. Using the aforementioned relation between FOS/C and random walks, we prove that in vertex-transitive graphs both parts must be connected components.

Cohesive zone representation and junction partitioning for crystal plasticity analyses

SUMMARYA novel scheme is presented for incorporating finite thickness cohesive interfaces in virtual grain structures for crystal plasticity finite element (CPFE) analyses of intergranular crack initiation and propagation. A Voronoi tessellation model is used to define the virtual grain structure, with automatically generated nonzero thickness cohesive zones (CZs) representing the grain boundaries and multiple junctions. An efficient grain boundary offsetting algorithm is presented, and issues related to automatically partitioning multiple junctions are discussed. Two feasible junction partitioning schemes are presented, the second of which has the advantage of partitioning junctions using uniform quadrilateral elements and naturally defining their normal and tangential directions. For the second scheme, a rule‐based method is presented that carries out the preliminary meshing of CZ junctions, including data representation, edge event processing, and cut and trim operations. A virtual grain structure modelling system, VGRAIN, is introduced to implement the proposed CZ junction partitioning method and directly generate meshed virtual grain structures with CZ grain boundaries for CPFE studies. To demonstrate the proposed junction partitioning and CZ representation schemes, two finite strain CPFE simulations are presented for plane strain uniaxial tension and three‐point bending, demonstrating large‐scale crack initiation and propagation under shear and opening modes. Copyright © 2012 John Wiley & Sons, Ltd.

On the Edge Connectivity of Direct Products with Dense Graphs

Let $${\kappa^{\prime}(G)}$$ be the edge connectivity of G and G × H the direct product of G and H. Let H be any graph with minimal degree $${\delta(H)>|V(H)|/2}$$ . We prove that for any graph $${G, \kappa^{\prime}(G\times H)=\textup{min}\{2\kappa^{\prime}(G)|E(H)|,\delta(G)\delta(H)\}}$$ . In addition, the structure of minimum edge cuts is described. As an application, we present a necessary and sufficient condition for G × K n (n ? 3) to be super edge connected.

On Edge Cut of Graphs Leaving Components of Order at Least Five

An edge cut of a connected graph is 5-restricted if its removal leaves every component having order at least five. Graphs that contain 5-restricted edge cuts are characterized in this paper. As a result, it is shown that a connected graph G of order at least 13 contains 5-restricted edge cuts if and only if $${G \setminus v}$$ contains a component of order at least five for every vertex v of graph G.

Flows and parity subgraphs of graphs with large odd-edge-connectivity

The odd-edge-connectivity of a graph G is the size of the smallest odd edge cut of G. Tutte conjectured that every odd-5-edge-connected graph admits a nowhere-zero 3-flow. As a weak version of this famous conjecture, Jaeger conjectured that there is an integer k such that every k-edge-connected graph admits a nowhere-zero 3-flow. Jaeger [F. Jaeger, Flows and generalized coloring theorems in graphs, J. Combin. Theory Ser. B 26 (1979) 205–216] proved that every 4-edge-connected graph admits a nowhere-zero 4-flow. Galluccio and Goddyn [A. Galluccio, L.A. Goddyn, The circular flow number of a 6-edge-connected graph is less than four, Combinatorica 22 (2002) 455–459] proved that the flow index of every 6-edge-connected graph is strictly less than 4. This result is further strengthened in this paper that the flow index of every odd-7-edge-connected graph is strictly less than 4. The second main result in this paper solves an open problem that every odd-(2k+1)-edge-connected graph contains k edge-disjoint parity subgraphs. The third main theorem of this paper proves that if the odd-edge-connectivity of a graph G is at least4⌈log2|V(G)|⌉+1, then G admits a nowhere-zero 3-flow. This result is a partial result to the weak 3-flow conjecture by Jaeger and improves an earlier result by Lai et al. The fourth main result of this paper proves that every odd-(4t+1)-edge-connected graph G has a circular(2t+1)even subgraph double cover. This result generalizes an earlier result of Jaeger.

Laser Cutting of Ceramic and Thermal Efficiency Analysis

Laser cutting of ceramics offers considerable advantages over the conventional cutting methods. Some of these advantages include precision of operation, high speed processing and low cost. In addition, cutting process is independent of the mechanical properties of the ceramic material such as hardness and fracture toughness. Alumina is one of the widely used ceramics in industry due to their resistance to harsh environments. Laser cutting of alumina requires high laser power due to high latent heat of melting and elevated melting temperature. In efficient laser cutting process, the low laser power is required. Consequently, for optimum laser cutting conditions of alumina tiles, the thermal efficiency of the cutting process may not be a maximum. The optimum laser cutting parameters provide crack free and parallel edge cuts. Therefore, investigation into laser cutting of alumina ties and the efficiency analysis becomes essential. In the present study, laser cutting of alumina tiles with 3 mm thickness is carried out and thermodynamic analysis associated with the efficiency analysis is introduced. The lump parameter method is incorporated in the thermodynamic analysis. The optimum cutting conditions are then related to the first and second law efficiencies.

Laser Cutting of Ceramic and Thermal Efficiency Analysis

Laser cutting of ceramics offers considerable advantages over the conventional cutting methods. Some of these advantages include precision of operation, high speed processing and low cost. In addition, cutting process is independent of the mechanical properties of the ceramic material such as hardness and fracture toughness. Alumina is one of the widely used ceramics in industry due to their resistance to harsh environments. Laser cutting of alumina requires high laser power due to high latent heat of melting and elevated melting temperature. In efficient laser cutting process, the low laser power is required. Consequently, for optimum laser cutting conditions of alumina tiles, the thermal efficiency of the cutting process may not be a maximum. The optimum laser cutting parameters provide crack free and parallel edge cuts. Therefore, investigation into laser cutting of alumina ties and the efficiency analysis becomes essential. In the present study, laser cutting of alumina tiles with 3 mm thickness is carried out and thermodynamic analysis associated with the efficiency analysis is introduced. The lump parameter method is incorporated in the thermodynamic analysis. The optimum cutting conditions are then related to the first and second law efficiencies.

Super restricted edge connectivity of regular edge-transitive graphs

An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−F contains an isolated edge for every minimum restricted edge cut F of G. It is proved in this paper that a connected regular edge-transitive graph of valency at least 3 is not super restricted edge connected if and only if it is either the three dimensional hypercube, or a tetravalent edge-transitive graph of girth 3 and of order at least 6. As a result, there are infinitely many k-regular Hamiltonian graphs with k≥3 which are not super restricted edge connected. This answers negatively a question in [J. Ou, F. Zhang, Super restricted edge connectivity of regular graphs, Graphs & Combin. 21 (2005) 459–467] regarding the relationship between restricted edge connected graphs and Hamiltonian graphs.

Optimally restricted edge connected elementary Harary graphs

An edge subset F of a connected graph G=(V,E) is a k-restricted edge cut if G−F is disconnected, and every component of G−F has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut. By the current studies on λk, it can be seen that the larger λk is, the more reliable the graph is. Hence one expects λk to be as large as possible. A possible upper bound for λk is ξk defined as ξk(G)=min{ω(S):0̸≠S⊂V(G),|S|=k and G[S] is connected }, where ω(S) is the number of edges with one end in S and the other end in V(G)∖S, and G[S] is the subgraph of G induced by S. A graph G is called λk-optimal if λk(G)=ξk(G). A natural question is whether there exists a graph G which is λk-optimal for any k≤|V(G)|/2. In this paper, we show that except for two cases, the elementary Harary graph has this property.

Edge fault tolerance of graphs with respect to super edge connectivity

A connected graph G is super edge connected (super-λ for short) if every minimum edge cut of G is the set of edges incident with some vertex. We define a super-λ graph G to be m-super-λ if G−S is still super-λ for any edge subset S with |S|⩽m. The maximum integer of such m, written as Sλ(G), is said to be the edge fault tolerance of G with respect to the super-λ property. In this paper, we study the bounds for Sλ(G), showing that min{λ′(G)−δ(G)−1,δ(G)−1}⩽Sλ(G)⩽δ(G)−1. More refined bounds are obtained for regular graphs and Cartesian product graphs. Exact values of Sλ are obtained for edge transitive graphs.

Edge fault tolerance of super edge connectivity for three families of interconnection networks

Let G = ( V, E) be a connected graph. G is said to be super edge connected (or super- λ for short) if every minimum edge cut of G isolates one of the vertex of G. A graph G is called m-super- λ if for any edge set S ⊆ E( G) with ∣ S∣ ⩽ m, G − S is still super- λ. The maximum cardinality of m-super- λ is called the edge fault tolerance of super edge connectivity of G. In this paper, we discuss the edge fault tolerance of super edge connectivity of three families of interconnection networks.

A neighborhood condition for graphs to be maximally k-restricted edge connected

For a connected graph G = ( V , E ) , an edge set S ⊆ E is a k-restricted edge cut 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 λ k ( G ) , is defined as the cardinality of a minimum k-restricted edge cut. Let ξ k ( G ) = min { | [ X , X ¯ ] | : | X | = k , G [ X ] is connected } . G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Let k ⩾ 4 be an integer. In this paper, we show that if | N G ( u ) ∩ N G ( v ) | ⩾ k for all pairs u , v of non-adjacent vertices and ξ k ( G ) ⩽ ⌊ | V | 2 ⌋ + k , then G is λ k -optimal.

Subwavelength imaging by a graded-index photonic-crystal flat lens in a honeycomb lattice

The left-handed behavior of a photonic-crystal flat lens with a graded index in a honeycomb lattice is proposed and theoretically studied. The performance of the flat superlens imaging of this structure has been demonstrated by finite-difference time-domain simulations. The full width at half-maximum of the image decreases to 62% compared to that of the image of a photonic-crystal slab without a graded index. The evanescent waves can be amplified and propagate to the far-field range. The image is not limited to be near the interface. The canalization effect of this structure is analyzed, and the tolerance of the edge cut of the graded-index structure is pretty good.

On super 2-restricted and 3-restricted edge-connected vertex transitive graphs

Let G = ( V ( G ) , E ( G ) ) be a simple connected graph and F ⊂ E ( G ) . An edge set F is an m -restricted edge cut if G − F is disconnected and each component of G − F contains at least m vertices. Let λ ( m ) ( G ) be the minimum size of all m -restricted edge cuts and ξ m ( G ) = min { | ω ( U ) | : | U | = m and G [ U ] is connected } , where ω ( U ) is the set of edges with exactly one end vertex in U and G [ U ] is the subgraph of G induced by U . A graph G is optimal- λ ( m ) if λ ( m ) ( G ) = ξ m ( G ) . An optimal- λ ( m ) graph is called super m -restricted edge-connected if every minimum m -restricted edge cut is ω ( U ) for some vertex set U with | U | = m and G [ U ] being connected. In this note, we give a characterization of super 2-restricted edge-connected vertex transitive graphs and obtain a sharp sufficient condition for an optimal- λ ( 3 ) vertex transitive graph to be super 3-restricted edge-connected. In particular, a complete characterization for an optimal- λ ( 2 ) minimal Cayley graph to be super 2-restricted edge-connected is obtained.

On a kind of reliability analysis of networks

A network is often modeled by a graph G=(V,E). An edge set F⊂E is a 3-restricted edge cut, if G−F is disconnected and every component of G−F has at least three vertices. The 3-restricted edge connectivity λ3(G) of G is the cardinality of a minimum 3-restricted edge cut of G. A graph G is λ3-connected, if 3-restricted edge cuts exist. A graph G is called λ3-optimal, if λ3(G)=ξ3(G), whereξ3(G)=min{|[U,U¯]|:U⊂V,|U|=3andG[U]is connected}.G[U] is the subgraph of G induced by the vertex subset U⊆V, and U¯=V⧹U is the complement of U·[U,U¯] is the set of edges with one end in U and the other in U¯. In this paper, we give some sufficient conditions for graphs to be λ3-optimal.

Super [formula omitted]-optimality of regular graphs

Let G = ( V , E ) be a connected graph. An edge set S ⊂ E is a 3-restricted edge cut, if G − S is disconnected and every component of G − S has at least three vertices. The 3-restricted edge connectivity λ 3 ( G ) of G is the cardinality of a minimum 3-restricted edge cut of G . A graph G is λ 3 -connected, if 3-restricted edge cuts exist. A graph G is called λ 3 -optimal, if λ 3 ( G ) = ξ 3 ( G ) , where ξ 3 ( G ) = m i n { | [ X , X ¯ ] | : X ⊆ V , | X | = 3 , G [ X ] i s c o n n e c t e d } , [ X , X ¯ ] is the set of edges of G with one end in X and the other in X ¯ and X ¯ = V − X . Furthermore, if every minimum 3-restricted edge cut is a set of edges incident to a connected subgraph induced by three vertices, then G is said to be super 3-restricted edge connected or super- λ 3 for simplicity. In this paper we show that let G be a k -regular connected graph of order n ≥ 6 , if k ≥ ⌊ n / 2 ⌋ + 3 , then G is super- λ 3 .

Operatives Management und Techniken der Mastektomie bei Frau-zu-Mann Transsexuellen

Subcutaneous mastectomy in female to male transsexuals is one first important step in gender adjustment. Beside proper surgical technique, operative management of mastectomy procedures is important to allow for an optimised process of gender transformation including all necessary individual operations. 126 female-to-male transsexuals were prospectively included in the observation period, all undergoing a step 1 operation of gender adjustment. The mastectomy procedure is conventionally embedded in an overall approach for gender transformation and was regularly combined with procedures like colpectomy, extension of the urethra as well as hysterectomy and ovariectomy. 4 different methods for mastectomy were introduced at our institution, depending on and adjusted to the breast volume, grade of breast ptosis, the breast size and envelope and skin elasticity. For patient evaluation a standardised survey was conducted using a semiquantitative score (1-4) for grading aesthetic results and nipple sensibility. Between 2006 and 2009 a total of 126 patients underwent a step 1 procedure of gender adjustment, 63 had mastectomy and 29 a revision procedure after external prior mastectomy. In 64/126 (50.8%) mastectomy was performed using a one-third edge cut, 30/126 (23.8%) were operated by a submammarian approach and pedicled nipples technique, in another 22/126 cases (17.5%) mastectomy was performed using circular tightening and 10/126 (7.9%) received freely transplanted nipples. For improving the aesthetic results, additional procedures were performed on 29 patients (46%). The patient survey revealed a high level of satisfaction with the aesthetic result, 42/50 (84%) of the patients rated results as good or very good. We newly introduced an operative concept of mastectomy being embedded and adjusted in an overall approach for gender adjustment (Munich Modular Operative Kit; MMOK). Using the presented individualised surgical techniques of mastectomy, all procedures were conducted in a very scar-saving manner, however, a higher rate of correction procedures was necessary to individually adjust skin shrinking and thereby optimising the prior aesthetic results. The new Munich Modular Operative Kit allows for gender reassignment including breast corrections usually using 4 operative steps.

Formula omitted]-optimally connected mixed Cayley graphs

A restricted edge cut of a graph X is an edge set whose removal disconnects X into non-trivial components. The cardinality of the minimum restricted edge cut is the restricted edge connectivity, denoted by λ ′ ( X ) . If X has restricted edge cuts and λ ′ ( X ) achieves the upper bound of the restricted edge connectivity, X is said to be λ ′ -optimal. In this work, we will prove that for all but a few exceptions, the mixed Cayley graph is λ ′ -optimal.

The Maximum Size of an Edge Cut and Graph Homomorphisms

For a graph G, let b(G)=max﹛|D|: Dis an edge cut of G﹜ . For graphs G and H, a map Ψ: V(G)→V(H) is a graph homomorphism if for each e=uv∈E(G), Ψ(u)Ψ(v)∈E(H). In 1979, Erdos proved by probabilistic methods that for p ≥ 2 with if there is a graph homomorphism from G onto Kp then b(G)≥f(p)|E(G)| In this paper, we obtained the best possible lower bounds of b(G) for graphs G with a graph homomorphism onto a Kneser graph or a circulant graph and we characterized the graphs G reaching the lower bounds when G is an edge maximal graph with a graph homomorphism onto a complete graph, or onto an odd cycle.

Fast Convergence of Markov Chain Monte Carlo Algorithms for Phylogenetic Reconstruction with Homogeneous Data on Closely Related Species

This paper studies a Markov chain for phylogenetic reconstruction which uses a popular transition between tree topologies known as subtree pruning-and-regrafting (SPR). We analyze the Markov chain in the simpler setting where the generating tree consists of very short edge lengths, short enough so that each sample from the generating tree (or character in phylogenetic terminology) is likely to have only one mutation, and where there are enough samples so that the data looks like the generating distribution. We prove in this setting that the Markov chain is rapidly mixing, i.e., it quickly converges to its stationary distribution, which is the posterior distribution over tree topologies. Our proofs use that the leading term of the maximum likelihood function of a tree is the maximum parsimony score, which is the size of the minimum cut in needed to realize single edge cuts of the generating tree. Our main contribution is a combinatorial proof that, in our simplified setting, SPR moves are guaranteed to converge quickly to the maximum parsimony tree. Our results are in contrast to recent works showing examples with heterogeneous data (namely, the data is generated from a mixture distribution) where many natural Markov chains are exponentially slow to converge to the stationary distribution.