Edge Cut Research Articles

Sufficient conditions for k-restricted edge connected graphs

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 λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min⁡{|[X,X¯]|:|X|=k,G[X]is connected}, where X¯=V\\X. G is maximally k-restricted edge connected (λk-optimal for short) if λk(G)=ξk(G). The k-restricted edge connectivity is more refined network reliability indices than edge connectivity. In this paper, let k≥2 be an integer, and let G be a graph of order ν(G) at least 2k satisfying |N(u)∩N(v)|≥2k−2 for all pairs u,v of nonadjacent vertices. If for each triangle T there exists at least one vertex v∈V(T) such that d(v)≥⌊ν(G)2⌋+k−1, then G is λk-optimal.

Homomorphisms of Signed Graphs

AbstractA signed graph is a graph G together with an assignment of signs + and − to all the edges of G where Σ is the set of negative edges. Furthermore and are considered to be equivalent if the symmetric difference of Σ1 and Σ2 is an edge cut of G. Naturally arising from matroid theory, several notions of graph theory, such as the theory of minors and the theory of nowhere‐zero flows, have been already extended to signed graphs. In an unpublished manuscript, B. Guenin introduced the notion of signed graph homomorphisms where he showed how some well‐known conjectures can be captured using this notion. A signed graph is said to map to if there is an equivalent signed graph of and a mapping such that (i) if then and if and only if . The chromatic number of a signed graph can then be defined as the smallest order of a homomorphic image of . Capturing the notion of graph homomorphism order, signed graph homomorphisms provide room for extensions and strengthenings of most homomorphism and coloring theories on graphs. Thus this paper is the first general study of signed graph homomorphisms. In this work, our focus would be on the relation of homomorphisms of signed graphs with minors of signed graphs. After a thorough introduction to the concept we show that the notion of signed graph homomorphism on the set of signed graphs whose underlying graph is bipartite already captures the standard notion of graph homomorphism. We prove that the largest planar signed clique is of order 8. For the maximum chromatic number of planar signed graphs we give the lower bound of 10 and the upper bound of 48. We determine this maximum for some other families such as outerplanar signed graphs. Finally, reformulating Hadwiger's conjecture in the language of homomorphism of signed graphs whose underlying graph is bipartite, we show that while some stronger form of the conjecture holds for small chromatic number, such strengthening of the conjecture would not hold for large chromatic numbers. This could be regarded as a first indication that perhaps Hadwiger's conjecture only holds for small chromatic numbers.

A Dependent Set Based Approach for Large Graph Analysis

Now a day’s social or computer networks produced graphs of thousands of nodes & millions of edges. Such Large graphs are used to store and represent information. As it is a complex data structure it requires extra processing. Partitioning or clustering methods are used to decompose a large graph. In this paper dependent set based graph partitioning approach is proposed which decomposes a large graph into sub graphs. It creates uniform partitions with very few edge cuts. It also prevents the loss of information. The work also focuses on an approach that handles dynamic updation in a large graph and represents a large graph in abstract form.

Effect of Rounding the Edges in the Base of a Rider on the Stress Intensity Factors in a Body with Edge Cracks

We study the dependence of the stress intensity factors in a body containing an edge arbitrarily oriented crack on the radius R of rounding of the edges in the base of a rider pressed with unilateral friction to the damaged body near the crack mouth. Within the framework of the two-dimensional problem, this body is modeled by an elastic half plane with edge cut and the action of the rider is replaced by a load reproducing the distribution of pressure established by Aleksandrov and Romalis [Contact Problems in Machine Building, Mashinostroenie, Moscow, 1986] in the contact zone of two elastic cylinders with initial contact strip. The numerical results are obtained for the identical materials of contacting bodies for a half length of the initial contact zone b = [5; 10] mm, the following values of the radius: R = [0.1; 1.0; 10.0; 100.0] mm, and different inclination angles of the edge crack, relative positions of the rider and the crack, and crack lengths.

Maximum genus, connectivity, and Nebeský's Theorem

We prove lower bounds on the maximum genus of a graph in terms of its connectivity and Betti number (cycle rank). These bounds are tight for all possible values of edge-connectivity and vertex-connectivity and for both simple and non-simple graphs. The use of Nebeský's characterization of maximum genus gives us both shorter proofs and a description of extremal graphs. An additional application of our method shows that the maximum genus is almost additive over the edge cuts.

Atoms of cyclic edge connectivity in regular graphs

A cyclic edge-cut of a connected graph $$G$$G is an edge set, the removal of which separates two cycles. If $$G$$G has a cyclic edge-cut, then it is called cyclically separable. For a cyclically separable graph $$G$$G, the cyclic edge connectivity of a graph $$G$$G, denoted by $$\lambda _c(G)$$?c(G), is the minimum cardinality over all cyclic edge cuts. Let $$X$$X be a non-empty proper subset of $$V(G)$$V(G). If $$[X,\overline{X}]=\{xy\in E(G)\ |\ x\in X, y\in \overline{X}\}$$[X,X¯]={xy?E(G)|x?X,y?X¯} is a minimum cyclic edge cut of $$G$$G, then $$X$$X is called a $$\lambda _c$$?c-fragment of $$G$$G. A $$\lambda _c$$?c-fragment with minimum cardinality is called a $$\lambda _c$$?c-atom. Let $$G$$G be a $$k (k\ge 3)$$k(k?3)-regular cyclically separable graph with $$\lambda _c(G)

A CACTUS THEOREM FOR END CUTS

Dinits–Karzanov–Lomonosov showed that it is possible to encode all minimal edge cuts of a graph by a tree-like structure called a cactus. We show here that minimal edge cuts separating ends of the graph rather than vertices can be "encoded" also by a cactus. As a corollary, we obtain a new proof of Stallings' ends theorem. We apply our methods to finite graphs as well and we show that several types of cuts can be encoded by cacti.

On the connectivity preserving minimum cut problem

In this paper, we study a generalization of the classical minimum cut problem, called Connectivity Preserving Minimum Cut (CPMC) problem, which seeks a minimum cut to separate a pair (or pairs) of source and destination nodes and meanwhile ensure the connectivity between the source and its partner node(s). For this problem, we consider two variants, connectivity preserving minimum node cut (CPMNC) and connectivity preserving minimum edge cut (CPMEC). For CPMNC, we show that it cannot be approximated within αlogn for some constant α unless P=NP, and cannot be approximated within any poly(logn) unless NP has quasi-polynomial time algorithms. The hardness results hold even for graphs with unit weight and bipartite graphs. For CPMEC, we show that it is in P for planar graphs.

Fault tolerance of hypercubes and folded hypercubes

Let $$G = (V,E)$$ G = ( V , E ) be a connected graph. The conditional edge connectivity $$\lambda _\delta ^k(G)$$ ? ? k ( G ) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects $$G$$ G and each component of $$G - F$$ G - F has $$\delta \ge k$$ ? ? k . We assume that $$F \subseteq E$$ F ⊆ E is an edge set, $$F$$ F is called edge extra-cut, if $$G - F$$ G - F is not connected and each component of $$G - F$$ G - F has more than $$k$$ k vertices. The edge extraconnectivity $$\lambda _\mathrm{e}^k(G)$$ ? e k ( G ) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.

On m-restricted edge connectivity of undirected generalized De Bruijn graphs

An m-restricted edge cut is an edge cut of a connected graph whose removal results in components of order at least m, the minimum cardinality over all m-restricted edge cuts of a graph is its m-restricted edge connectivity. It is known that telecommunication networks with topology having larger m-restricted edge connectivity are locally more reliable for all m≤3. This work shows that if n≥7, then undirected generalized binary De Bruijn graph UBG(2, n) is maximally m-restricted edge connected for all m≤3, where a graph G is maximally m-restricted edge connected if its m-restricted edge connectivity is equal to the minimum number of edges from any connected subgraphs S to G−S.

On the Z‐eigenvalues of the signless Laplacian tensor for an even uniform hypergraph

SUMMARYWe generalize the signless Laplacian matrices for graphs to the signless Laplacian tensors for even uniform hypergraphs and set some fundamental properties for the spectral hypergraph theory based upon the signless Laplacian tensors. In particular, the smallest and the largest Z‐eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are studied, and as an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving these two Z‐eigenvalues are presented. Copyright © 2013 John Wiley & Sons, Ltd.

Nowhere-zero 3-flows of graphs with prescribed sizes of odd edge cuts

The relation between sizes of odd-cycles and chromatic numbers of graphs has been extensively studied. It was conjectured by Bollobás and Erdős (1990) [4] and proved by Gyárfás [A. Gyárfás, Graph with k odd cycle lengths, Discrete Math. 103 (1992) 41–48] that if a graphGhaskdifferent odd cycle sizes, then the chromatic number ofG, χ(G)≤2k+2. Recently, Camacho and Schiermeyer [S.M. Camacho, I. Schiermeyer, Colorings of graphs with two consecutive odd cycle lengths, Discrete Math. 309 (2009) 4916–4919] proved that every graph containing only odd cycles of lengths2k−1and2k+1is 4-colorable. Wang [S.S. Wang, Structure and coloring of graphs with only small odd cycles, SIAM J. Discrete Math. 22 (2008) 1040–1072] characterizes all 3-colorable graphs whose only odd cycles are 3-cycles and 5-cycles; and Kaiser et al. [T. Kaiser, O. Rucký, R. Škrekovski, Graphs with odd cycle lengths 5 and 7 are 3-colorable, SIAM J. Discrete Math. 25 (2012) 1069–1088] proved that every graph with odd cycle lengths 5 and 7 is 3-colorable.In this paper, we investigate the dual version of this problem: the relation between integer flows and odd edge cuts (the integer flow problem is a dual version of the vertex coloring problem, and the dual of an odd-cycle is an odd-edge-cut). The main result in this paper shows that, for each odd integer k≥3, if the size of every odd edge cut of a graph G is in {k,k+2,…,3k−2}, then G admits a nowhere-zero 3-flow if and only if G cannot be reduced to three well-defined families of graphs.

A sufficient condition for graphs to be [formula omitted]-optimal

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 λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X] is connected}. G is λk-optimal if λk(G)=ξk(G). In 2004, Hellwig and Volkmann gave a sufficient condition for λ2-optimality in graphs of diameter 2. In this paper, we extend the result and give a similar sufficient condition for λk-optimality in graphs of diameter 2 with k≥3.

Damages of wind turbine blade trailing edge: Forms, location, and root causes

The geometrical form and the manufacturing technique make the trailing edge of the wind turbine blade more susceptible to damage. In this study the trailing edge in a number of 81blades of 100kW wind turbines and 18blades of 300kW wind turbines of working life ranging between 6.5×107and 1.1×108cycles were completely visually scanned. The different damages were classified and allocated in their exact position relative to the blade length. Cracks in different orientations with the blade length were the frequent types of damages which found on the trailing edge. First, longitudinal cracks (LCs) that found along the blade trailing edge from the blade root to the tip were in lengths that varied from a few centimeters up to around 1.35m. Second, transverse cracks (TCs) were found in either simple TCs which growing on one shell at the trailing edge or round TCs which growing across the two shells. TCs lengths were ranging from 20 to 50mm. Third, edge damages were detected in the form of edge cuts or crushing. The possible root causes of the different types of cracks have been discussed.

Crossing number additivity over edge cuts

Consider a graph G with a minimal edge cut F and let G1, G2 be the two (augmented) components of G−F. A long-open question asks under which conditions the crossing number of G is (greater than or) equal to the sum of the crossing numbers of G1 and G2—which would allow us to consider those graphs separately. It is known that crossing number is additive for |F|∈{0,1,2} and that there exist graphs violating this property with |F|≥4. In this paper, we show that crossing number is additive for |F|=3, thus closing the final gap in the question.The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing-critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.

H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph

The signless Laplacian tensor and its H-eigenvalues for an even uniform hypergraph are introduced in this paper. Some fundamental properties of them for an even uniform hypergraph are obtained. In particular, the smallest and the largest H-eigenvalues of the signless Laplacian tensor for an even uniform hypergraph are discussed, and their relationships to hypergraph bipartition, minimum degree, and maximum degree are described. As an application, the bounds of the edge cut and the edge connectivity of the hypergraph involving the smallest and the largest H-eigenvalues are presented.

High-dimensional clustering: a clique-based hypergraph partitioning framework

Hypergraph partitioning has been considered as a promising method to address the challenges of high-dimensional clustering. With objects modeled as vertices and the relationship among objects captured by the hyperedges, the goal of graph partitioning is to minimize the edge cut. Therefore, the definition of hyperedges is vital to the clustering performance. While several definitions of hyperedges have been proposed, a systematic understanding of desired characteristics of hyperedges is still missing. To that end, in this paper, we first provide a unified clique perspective of the definition of hyperedges, which serves as a guide to define hyperedges. With this perspective, based on the concepts of shared (reverse) nearest neighbors, we propose two new types of clique hyperedges and analyze their properties regarding purity and size issues. Finally, we present an extensive evaluation using real-world document datasets. The experimental results show that, with shared (reverse) nearest neighbor-based hyperedges, the clustering performance can be improved significantly in terms of various external validation measures without the need for fine tuning of parameters.

Finding 2-Factors Closer to TSP Tours in Cubic Graphs

In this paper we are interested in algorithms for finding $2$-factors that cover certain prescribed edge-cuts in bridgeless cubic graphs. Since a Hamilton cycle is a 2-factor covering all edge-cuts, imposing the constraint of covering those edge-cuts makes the obtained $2$-factor closer to a Hamilton cycle. We present an algorithm for finding a minimum-weight $2$-factor covering all the $3$-edge cuts in weighted bridgeless cubic graphs, together with a polyhedral description of such 2-factors and that of perfect matchings intersecting all the 3-edge cuts in exactly one edge. We further give an algorithm for finding a 2-factor covering all the $3$- and $4$-edge cuts in bridgeless cubic graphs. Both of these algorithms run in ${\rm O}(n\sp{3})$ time, where $n$ is the number of vertices. As an application of the latter algorithm, we design a 6/5-approximation algorithm for finding a minimum 2-edge-connected spanning subgraph in 3-edge-connected cubic graphs, which improves upon the previous best ratio of 5/4...

A bubble-inspired algorithm for finite element mesh partitioning

SUMMARY This paper presents a bubble-inspired algorithm for partitioning finite element mesh into subdomains. Differing from previous diffusion BUBBLE and Center-oriented Bubble methods, the newly proposed algorithm employs the physics of real bubbles, including nucleation, spherical growth, bubble–bubble collision, reaching critical state, and the final competing growth. The realization of foaming process of real bubbles in the algorithm enables us to create partitions with good shape without having to specify large number of artificial controls. The minimum edge cut is simply achieved by increasing the volume of each bubble in the most energy efficient way. Moreover, the order, in which an element is gathered into a bubble, delivers the minimum number of surface cells at every gathering step; thus, the optimal numbering of elements in each subdomain has naturally achieved. Because finite element solvers, such as multifrontal method, must loop over all elements in the local subdomain condensation phase and the global interface solution phase, these two features have a huge payback in terms of solver efficiency. Experiments have been conducted on various structured and unstructured meshes. The obtained results are consistently better than the classical kMetis library in terms of the edge cut, partition shape, and partition connectivity. Copyright © 2012 John Wiley & Sons, Ltd.

Onr-Connected Graphs with No Semi-topologicalr-Wheel

A semi-topological r-wheel, denoted by , is a subdivision of the r-wheel preserving the spokes. This paper describes the r-connected graphs having no -subgraphs. For , these are shown to be only , while the class of 3-connected -free graphs is unexpectedly rich. First, every graph G in has an efficiently recognizable set of “contractible edges” (sometimes empty) such that a contraction minor belongs to if and only if F is a part of this set. So, the subclass of ante-contraction members of plays a key role. Second, the members of have 3-edge cuts. The familiar cactus representation of minimum edge cuts (Dinits et al., in: Issledovaniya po Diskretnoy Optimizatsii, Nauka, Moscow, pp. 290–306, 1976 (Russian); also A. Schrijver, Combinatorial Optimization (Polyhedra and Efficiency), Algorithms and Combinatorics, Vol. 24, Springer, 2003, p. 253) maps onto the class of trees whose internal vertices have even degrees, equal to 6 for any vertex adjacent to a leaf. The description of (quite concise as expressed in appropriate terms) refers to the explicit reconstruction of the reverse image of such a tree. We also derive the upper bound on the number of edges in an arbitrary n-vertex -free graph, , and conjecture that its maximum equals .