Minimum Edge Cut Research Articles

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.)

On Cyclic Edge-Connectivity and Super-Cyclic Edge-Connectivity of Double-Orbit Graphs

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is said to be cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity \(\lambda _c(G)\) is the cardinality of a minimum cyclic edge-cut of G. Let \(\zeta (G)=\min \{\omega (X) | X \text{ induce } \text{ a } \text{ shortest } \text{ cycle } \text{ in } G\}\), where \(\omega (X)\) is the number of edges with one end in X and the other end in \(V(G)-X\). A cyclically separable graph G with \(\lambda _c(G)=\zeta (G)\) is said to be cyclically optimal. In particular, we call a graph G super cyclically edge-connected if every minimum cyclic edge-cut isolates a shortest cycle of G. In this work, we first discuss the cyclic edge-connectivity of vertex transitive graphs, regular double-orbit graphs, and the double-orbit graphs with two orbits of same size; moreover, we also discuss the super-cyclic edge-connectivity of double-orbit graphs mentioned above.

Symmetric Property and Reliability of Balanced Hypercube

Huang and Wu in [IEEE Transactions on Computers 46 (1997) 484-490] introduced the balanced hypercube $BH_n$ as an interconnection network topology for computing systems, and they proved that $BH_n$ is vertex-transitive. However, some other symmetric properties, say edge-transitivity and arc-transitivity, of $BH_n$ remained unknown. In this paper, we solve this problem and prove that $BH_n$ is an arc-transitive Cayley graph. Using this, we also investigate some reliability measures, including super-connectivity, cyclic connectivity, etc., in $BH_n$ . First, we prove that every minimum edge-cut of $BH_n (n\ge 2)$ isolates a vertex, and every minimum vertex-cut of $BH_n (n\ge 3)$ isolates a vertex. This is stronger than that obtained by Wu and Huang which shows the connectivity and edge-connectivity of $BH_n$ are $2n$ . Second, Yang [Applied Mathematics and Computation 219 (2012) 970-975.] proved that for $n\ge 2$ , the super-connectivity of $BH_n$ is $4n-4$ and the super edge-connectivity of $BH_n$ is $4n-2$ . In this paper, we proved that $BH_n (n\ge 2)$ is super- $\lambda^{\prime }$ but not super- $\kappa^{\prime }$ . That is, every minimum super edge-cut of $BH_n (n\ge 2)$ isolates an edge, but the minimum super vertex-cut of $BH_n (n\ge 2)$ does not isolate an edge. Third, we also obtain that for $n\ge 2$ , the cyclic connectivity of $BH_n$ is $4n-4$ and the cyclic edge-connectivity of $BH_n$ is $4(2n-2)$ . That is, to become a disconnected graph which has at least two components containing cycles, we need to remove at least $4n-4$ vertices (resp. $4(4n-2)$ edges) from $BH_n (n\ge 2)$ .

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)

Super-edge-connectivity of G( k, d, s)(s ⩾ k/2)

A graph G is super-edge-connected, for short super-λ, if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree. Alphabet overlap graph G(k, d, s) is undirected, simple graph with vertex set \(V = \left\{ {\left. v \right|v = \left( {v_1 \cdots v_k } \right);v_i \in \left\{ {1,2, \cdots ,d} \right\},i = 1, \cdots ,k} \right\}\). Two vertices \(u = \left( {u_1 \cdots u_k } \right)\) and \(v = \left( {v_1 \cdots v_k } \right)\) are adjacent if and only if us+i =vi or \(v_{s + i} = u_i \left( {i = 1, \cdots ,k - s} \right)\). In particular G(k, d, 1) is just an undirected de Bruijn graph. In this paper, we show that the diameter of G(k, d, s) is \(\left\lceil {\frac{k} {s}} \right\rceil\), the girth is 3. Finally, we prove that G(k, d, s)(s ⩾ k /2) is super-λ.

On the existence of super edge-connected graphs with prescribed degrees

Let G be a connected graph of order n, minimum degree δ(G), and edge-connectivity κ′(G). The graph G is maximally edge-connected if κ′(G)=δ(G) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree.A list (d1,…,dn) is graphic if there is a graph with vertices v1,…,vn such that d(vi)=di for 1≤i≤n. A graphic list D is super edge-connected if D is the degree list of some super edge-connected graph. We prove that a graphic list D with least element 1 is super edge-connected if and only if (1) ∑i=1ndi≥2n or (2) ∑i=1ndi=2(n−1) and max{di:1≤i≤n}=n−1. We also give a necessary and sufficient condition for a graphic list with least entry 2 to be super edge-connected, and we show that every graphic list with least element at least 3 is super edge-connected.

Super s-restricted edge-connectivity of vertex-transitive graphs

Let G be a connected graph with vertex-set V (G) and edge-set E(G). A subset F of E(G) is an s-restricted edge-cut of G if G - F is disconnected and every component of G - F has at least s vertices. Let λ s (G) be the minimum size of all s-restricted edge-cuts of G and ξ s (G)=min{|[X, V (G)\X]|: |X| = s, G[X] is connected}, where [X, V (G)\X] is the set of edges with exactly one end in X. A graph G with an s-restricted edge-cut is called super s-restricted edge-connected, in short super-λ s , if λ s (G) = ξ s (G) and every minimum s-restricted edge-cut of G isolates one component G[X] with |X| = s. It is proved in this paper that a connected vertex-transitive graph G with degree k > 5 and girth g > 5 is super-λ s for any positive integer s with s ⩽ 2g or s ⩾ 10 if k = g = 6.

On restricted edge-connectivity of vertex-transitive multigraphs

Let G=(V, E) be a multigraph. The multigraph G is called maximally edge-connected if λ(G)=δ(G), and super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edge-connectivity λ′(G) of G is the minimum number of edges whose removal disconnects G into non-trivial components. If λ′(G) achieves the upper bound of restricted edge-connectivity, then G is said to be λ′-optimal. This work characterizes maximally edge-connected vertex-transitive multigraphs, super edge-connected vertex-transitive multigraphs, and λ′-optimal vertex-transitive multigraphs.

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.

The [formula omitted]-chromatic number of regular graphs via the edge-connectivity

The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a proper vertex coloring with k colors in which each color class contains a vertex that is adjacent to at least one vertex in each of the other color classes. El Sahili and Kouider asked whether φ(G)=d+1 for every d-regular graph G with girth at least 5. Blidia, Maffray, and Zemir showed that the Petersen graph provides a negative answer to this question and conjectured that the Petersen graph is the only exception. In this note, we investigate a strengthened form of the question.A d-regular graph G is super-edge-connected if every minimum edge-cut is the set of all edges incident to one vertex in G, i.e., the edge-connectivity of G is equal to d and deleting every minimum edge-cut of G isolates a vertex. We show that if G is a d-regular graph that contains no 4-cycle as a subgraph, then φ(G)=d+1 whenever G is not super-edge-connected.

A characterization of the edge connectivity of direct products of graphs

The direct product of graphs G=(V(G),E(G)) and H=(V(H),E(H)) is the graph, denoted as G×H, with vertex set V(G×H)=V(G)×V(H), where vertices (x1,y1) and (x2,y2) are adjacent in G×H if x1x2∈E(G) and y1y2∈E(H). The edge connectivity of a graph G, denoted as λ(G), is the size of a minimum edge-cut in G. We introduce a function ψ and prove the following formula λ(G×H)=min{2λ(G)|E(H)|,2λ(H)|E(G)|,δ(G×H),ψ(G,H),ψ(H,G)}. We also describe the structure of every minimum edge-cut in G×H.

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 .

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.

Inverse degree and super edge-connectivity

Let G be a connected graph of order n, minimum degree δ(G) and edge connectivity λ(G). The graph G is called maximally edge-connected if λ(G)=δ(G), and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. Define the inverse degree of G with no isolated vertices as R(G)=∑ v∈V(G)1/d(v), where d(v) denotes the degree of the vertex v. We show that if R(G)<2+(n−2δ)/(n−δ) (n−δ−1), then G is super edge-connected. We also give an analogous result for triangle-free graphs.

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.

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 2-Approximation for the Minimum Duplication Speciation Problem

We consider the following problem: given a set of gene family trees, spanning a given set of species, find a first speciation which splits these species into two subsets and minimizes the number of gene duplications that happened before this speciation. We call this problem the Minimum Duplication Bipartition Problem. Using a generalization of the Minimum Edge-Cut Problem, we propose a polynomial time 2-approximation algorithm for the Minimum Duplication Bipartition Problem. We apply this algorithm to the inference of species trees on synthetic datasets and on two datasets of eukaryotic species.