Edge Cut Research Articles

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.

On Mergings in Acyclic Directed Graphs

Consider an acyclic directed graph $G$ with sources $s_1, s_2, \ldots,s_n$ and sinks $r_1, r_2, \ldots, r_n$. For $i=1, 2, \ldots,n$, let $c_i$ denote the size of the minimum edge cut between $s_i$...

Domination game and minimal edge cuts

In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular, if C a minimum edge cut of a connected graph G, then γg(G)≤γg(G∖C)+2κ′(G). Double-Staller graphs are introduced in order to show that this upper bound can be attained for graphs with a bridge. The obtained results are used to extend the family of known traceable graphs whose game domination numbers are at most one-half their order. Along the way two technical lemmas, which seem to be generally applicable for the study of the domination game, are proved.

An Investigation of Damages in Low Power Wind Turbine Blades

Data in the literature about the various types of damages in wind turbines which may result during manufacturing or operation is not enough. In the current work, a number of 18 blades of 300 kW and 81 blades of 100 kW power wind turbines are inspected using visual test (VT) and if needed using liquid penetrant test (PT) when needed. The location and size of discontinuities found are measured and documented. Types of discontinuities are categorized into four main ‎groups: 1) Transverse cracks and longitudinal cracks, 2) Hair line surface cracks, coating pores, surface damages, holes or ‎penetrations, 3) Edge cuts or crushing ‎and side separation, and 4) reworked areas.‎ The gathered data are analyzed to allocate the different forms of discontinuities about the blade length, to establish an indication of the rotor blades working conditions. It is found that the transverse carks are concentrated at the highly loaded and geometric change region of trailing edge. Longitudinal cracks up to 59 cm long are detected also at the region of the geometric change; i.e. in the root and the cover of the aerodynamic zone. Edge damages such as edge cuts or crushing and partial side separation are found in the sharp trailing edge. Surface and coating damages are observed on the leading edge due to erosion.

Vulnerability of super connected split graphs and bisplit graphs

A graph $G = (C, I, E)$ is called a split graph if its vertex set $V$ can be partitioned into a clique $C$ and an independent set $I$. A graph $G = (Y \cup Z, I, E)$ is called a bisplit graph if its vertex set $V$ can be partitioned into three stable sets $I, Y,Z$ such that $Y \cup Z$ induces a complete bipartite graph and an independent set $I$. A connected graph $G$ is called supper-$κ$ (resp. super-$λ$) if every minimum vertex cut (edge cut) of $G$ is the set of neighbors of some vertex (the edges of incident to some vertex) in $G$. In this note, we show that: split graphs and bisplit graphs are super-$κ$ and super-$λ$.

A complexity dichotomy for matching cut in (bipartite) graphs of fixed diameter

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 even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer d≥3, Matching Cut is NP-complete in the class of graphs of diameter d. This resolves an open problem posed by Borowiecki and Jesse-Józefczyk (2008) [6].We then show that, for any fixed integer d≥4, Matching Cut is NP-complete even when restricted to the class of bipartite graphs of diameter d. Complementing the hardness results, we show that Matching Cut is polynomial-time solvable in the class of bipartite graphs of diameter at most three, and point out a new and simple polynomial-time algorithm solving Matching Cut in graphs of diameter 2.

Sufficient conditions for graphs with girth [formula omitted] to be maximally [formula omitted]-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 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}, where X̄=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 index than the edge connectivity. In this paper, for graph G with girth g≥2k+1 and minimum degree δ≥k, we present a lower bound on the cardinality of λk-atoms of G when λk(G)<ξk(G). Moreover, we give some sufficient conditions for graphs with girth g to be λk-optimal.

General Transmission Lemma and Wiener complexity of triangular grids

The Transmission Lemma from Rajasingh et al. (2016) is extended to the General Transmission Lemma. It gives a formula for the transmission of a vertex u as a function of a collection of edge cuts and a u-routing that uniformly intersects the edge cuts. The applicability of the General Transmission Lemma is demonstrated by computing the Wiener complexity of triangular grid networks.

Graph Bisection with Pareto Optimization

We introduce FlowCutter, a novel algorithm to compute a set of edge cuts or node separators that optimize cut size and balance in the Pareto sense. Our core algorithm heuristically solves the balanced connected st -edge-cut problem, where two given nodes s and t must be separated by removing edges to obtain two connected parts. Using the core algorithm as a subroutine, we build variants that compute node separators that are independent of s and t . From the computed Pareto set, we can identify cuts with a particularly good tradeoff between cut size and balance that can be used to compute contraction and minimum fill-in orders, which can be used in Customizable Contraction Hierarchies (CCHs), a speed-up technique for shortest-path computations. Our core algorithm runs in O ( c ∣ E ∣) time, where E is the set of edges and c is the size of the largest outputted cut. This makes it well suited for separating large graphs with small cuts, such as road graphs, which is the primary application motivating our research. For road graphs, we present an extensive experimental study demonstrating that FlowCutter outperforms the current state of the art in terms of both cut sizes and CCH performance. By evaluating FlowCutter on a standard graph partitioning benchmark, we further show that FlowCutter also finds small, balanced cuts on nonroad graphs. Another application is the computation of small tree decompositions. To evaluate the quality of our algorithm in this context, we entered the PACE 2016 challenge [13] and won first place in the corresponding sequential competition track. We can therefore conclude that our FlowCutter algorithm finds small, balanced cuts on a wide variety of graphs.

Streaming Algorithms for News and Scientific Literature Recommendation: Monotone Submodular Maximization With a &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$d$ &lt;/tex-math&gt; &lt;/inline-formula&gt;-Knapsack Constraint

Submodular optimization plays a significant role in combinatorial problems, since it captures the structure of the edge cuts in graphs, the coverage of sets, and so on. Many data mining and machine learning problems can be cast as submodular maximization problems with applications in recommendation systems and data diversification. In this paper, we focus on the problem of maximizing a monotone submodular function subject to a $d$ -knapsack constraint, for which we propose a streaming algorithm that achieves a $(({1}/{1+2d})-\epsilon)$ -approximation of the optimal value, while it only needs one single pass through the data set without storing all the data in the memory. In our experiments, we extensively evaluate the effectiveness of our proposed algorithm via two applications: news recommendation and scientific literature recommendation. It is observed that the proposed streaming algorithm achieves both execution speedup and memory saving by several orders of magnitude, compared with existing approaches.

Reliability Analysis of Crossed Cube Networks on Degree

Crossed cubes network is a kind of interconnection structure as a basis for distributed memory parallel computer architecture. Reliability takes an important role in fault tolerant computing on multiprocessor systems. Connectivity is a vital metric to explore fault tolerance and reliability of network structure based on a graph model. Let be a connected graph. The k-conditional edge connectivity is the cardinality of the minimum edge cuts , if any, whose deletion disconnects and each component of has property of minimum degree . The k-conditional connectivity can be defined similarly. In this paper, we determine the k- conditional (edge) connectivity of crossed cubes for small k. And we also prove other properties of .

Rules for folding polyminoes from one level to two levels

Abstract Polyominoes have been the focus of many recreational and research investigations. In this article, the authors investigate whether a paper cutout of a polyomino can be folded to produce a second polyomino in the same shape as the original, but now with two layers of paper. For the folding, only “corner folds” and “half edge cuts” are allowed, unless the polyomino forms a closed loop, in which case one is allowed to completely cut two squares in the polyomino apart. With this set of allowable moves, the authors present algorithms for folding different types of polyominoes and prove that certain polyominoes can successfully be folded to two layers. The authors also establish that other polyominoes cannot be folded to two layers if only these moves are allowed.

New algorithms for the minimum coloring cut problem

AbstractThe minimum coloring cut problem is defined as follows: given a connected graph G with colored edges, find an edge cut E' of G (a minimal set of edges whose removal renders the graph disconnected) such that the number of colors used by the edges in E' is minimum. In this work, we present two approaches based on variable neighborhood search to solve this problem. Our algorithms are able to find all the optimum solutions described in the literature.

Multigraphs without large bonds are wqo by contraction

AbstractWe show that the class of multigraphs with at most p connected components and bonds of size at most k is well‐quasi‐ordered by edge contraction for all positive integers . (A bond is a minimal nonempty edge cut.) We also characterize canonical antichains for this relation.

Reliability analysis of twisted cubes

Connectivity is a vital metric to explore fault tolerance and reliability of network structure based on a graph model. Let G=(V,E) be a connected graph. A connected graph G is called supper-κ (resp. super-λ) if every minimum vertex cut (edge cut) of G is the set of neighbors of some vertex in G. Let F⊆V be a vertex set, F is called extra-cut, if G−F is not connected and each component of G−F has more than k vertices. The extraconnectivity κk(G) is the cardinality of the minimum extra-cuts. A r-component cut of G is a set S of vertices, G−S has at least r components. r-component connectivity cκr(G) of G is the size of the smallest r-component cut. The r-component edge connectivity cλr(G) can be defined similarly. In this paper, we determine the r-component (edge) connectivity of twisted cubes TNn for small r. And we also prove other properties of TNn.

Minimum Cuts of Distance-Regular Digraphs

In this paper, we investigate the structure of minimum vertex and edge cuts of distance-regular digraphs. We show that each distance-regular digraph $\Gamma$, different from an undirected cycle, is super edge-connected, that is, any minimum edge cut of $\Gamma$ is the set of all edges going into (or coming out of) a single vertex. Moreover, we will show that except for undirected cycles, any distance regular-digraph $\Gamma$ with diameter $D=2$, degree $k\leq 3$ or $\lambda=0$ ($\lambda$ is the number of 2-paths from $u$ to $v$ for an edge $uv$ of $\Gamma$) is super vertex-connected, that is, any minimum vertex cut of $\Gamma$ is the set of all out-neighbors (or in-neighbors) of a single vertex in $\Gamma$. These results extend the same known results for the undirected case with quite different proofs.

ON SURFACE FRACTURE OF RAIL HEADS

Purpose. The formation of crack-like defects in rails of railway tracks is a serious problem for engineering practice because of the danger of creating emergency situations. The purpose of this work is to establish theoretically the characteristic angle of propagation of surface cracks in the rail heads of railway rails, which is basic in the formation of typical surface contact fatigue damages, such as pitting, «checks» and «squat». It is also necessary to find the conditions for determining this angle. Methodology. The investigations were carried out on the basis of the method of singular integral equations. The rail damaged by the surface crack was modeled with a half-plane with an edge cut, and the action of the wheel on the rail by unidirectional repeated translational movement along the edge of the half-plane of the Hertzian contact forces with the tangential component. The problem of determining the stress intensity factors in the vicinity of the crack tip in the rail head was reduced to a system of two real singular integral equations which were solved numerically by the Gauss-Chebyshev mechanical quadrature method. The complexity of the problem consists in the fact that the boundaries of the contact areas and the opening of the crack faces are unknown beforehand and they change when the model contact forces move. These boundaries were determined simultaneously with solving the integral equations of the problem from additional conditions by the iteration method. Findings. The presence of the characteristic angle of propagation of mode II surface cracks in the rail head has been established theoretically and the conditions for its determination have been put down. The results obtained are in good agreement with engineering and experimental data. Originality. For the first time, the values of the characteristic angle were theoretically determined, under which at the initial stage, the surface contact fatigue cracks propagate in the head of the railway rail under the action of the wheels. Conditions for determining this angle have been also put down. Practical value. The received data are of great importance for engineering practice, since they reveal the nature of surface contact fatigue defects under various operating conditions and allow to predict their contact strength and durability.

Improved bounds and algorithms for graph cuts and network reliability

AbstractKarger (SIAM Journal on Computing, 1999) developed the first fully‐polynomial approximation scheme to estimate the probability that a graph G becomes disconnected, given that its edges are removed independently with probability p. This algorithm runs in time to obtain an estimate within relative error . We improve this run‐time through algorithmic and graph‐theoretic advances. First, there is a certain key sub‐problem encountered by Karger, for which a generic estimation procedure is employed; we show that this has a special structure for which a much more efficient algorithm can be used. Second, we show better bounds on the number of edge cuts which are likely to fail. Here, Karger's analysis uses a variety of bounds for various graph parameters; we show that these bounds cannot be simultaneously tight. We describe a new graph parameter, which simultaneously influences all the bounds used by Karger, and obtain much tighter estimates of the cut structure of G. These techniques allow us to improve the runtime to ; our results also rigorously prove certain experimental observations of Karger and Tai (Proc. ACM‐SIAM Symposium on Discrete Algorithms, 1997). Our rigorous proofs are motivated by certain non‐rigorous differential‐equation approximations which, however, provably track the worst‐case trajectories of the relevant parameters. A key driver of Karger's approach (and other cut‐related results) is a bound on the number of small cuts: we improve these estimates when the min‐cut size is “small” and odd, augmenting, in part, a result of Bixby (Bulletin of the AMS, 1974).

Compatible Cycle Decomposition of bad K5-minor-free graphs

Let G be an eulerian graph. For each vertex v∈V(G), let T(v) be a non-empty subset of a partition of the edges incident with v into 2-subsets and set T=∪v∈V(G)T(v), called a transition system of G. A transition system T of G is admissible if T∩F|≤12|F| for every T∈T and every edge cut F of G. A cycle decomposition C of G is called a compatible cycle decomposition (CCD for short) of (G, T) if |E(C)∩T|≤1 for every cycle C∈C and every T∈T. H. Fleischner proved that if G is planar, then for every admissible transition system T of G, (G, T) has a CCD. G. Fan and C.-Q. Zhang (2000, J. Combin. Theory Ser. B 78, 1-23) showed that this result is also true for K5-minor-free graphs. We generalize this result to all eulerian graphs that do not contain a special type of K5-minor which is called a bad K5-minor. To this purpose, we characterise the 4−regular “hereditary” bad K5-minor-free graphs (G, T) in which every CCD of (G, T) is a pair of hamiltonian cycles.

Finding a maximum 2-matching excluding prescribed cycles in bipartite graphs

We introduce a new framework for restricted 2-matchings close to Hamilton cycles. For an undirected graph (V,E) and a family U of vertex subsets, a 2-matching F is called U-feasible if |F[U]|≤|U|−1 for each U∈U, where F[U] is the set of edges in F induced by U. If F is a 2-factor, this property makes F satisfy the subtour elimination constraint for U∈U. Our framework can describe C≤k-free 2-matchings, i.e., 2-matchings without cycles of at most k edges, and 2-factors covering prescribed edge cuts, both of which are intensively studied as relaxations of Hamilton cycles. While the problem of finding a maximum U-feasible 2-matching is NP-hard, we prove that the problem is tractable when the graph is bipartite and each U∈U induces a Hamilton-laceable graph. This case generalizes the C≤4-free 2-matching problem in bipartite graphs. We establish a min-max theorem, a combinatorial polynomial-time algorithm, and decomposition theorems by extending the theory of C≤4-free 2-matchings. Our result provides the first polynomially solvable case for the maximum C≤k-free 2-matching problem for k≥5. For instance, in bipartite graphs in which every cycle of length six has at least two chords, our algorithm solves the maximum C≤6-free 2-matching problem in O(n2m) time, where n and m are the numbers of vertices and edges, respectively.