We describe a new circulation-based method to determine cuts in an undirected graph. A circulation is an oriented labeling of edges with integers so that at each vertex, the sum of the in-labels equals the sum of out-labels. For an integer k, our approach is based on simple algorithms for sampling a circulation (mod k) uniformly at random. We prove that with high probability, certain dependenci...

An r-cut of the complete r-uniform hypergraph Kr n is obtained by partitioning its vertex set into r parts and taking all edges that meet every part in exactly one vertex. In other words it is the edge set of a spanning complete r-partite subhypergraph of Kr n. An r-cut cover is a collection of r-cuts so that each edge of K r n is in at least one of the cuts. While in the graph case r = 2 any 2...

1. Let G = (V,E) be a c-edge connected graph. In other words, assume that the size of minimum cut in G is at least c. Construct a graph G′(V,E ′) by sampling each edge of G with probability p independently at random and reweighing each edge with weight 1/p. Suppose c > log n, and is such that log(n) c 2 ≤ 1. Show that as long as p ≥ log(n) c 2 , with high probability the size of every cut in G′...

The maximum flow algorithm has long been known for calculating the minimum edge cut of any two vertices of a connected graph. The original algorithm, however, does not tell us which edges should be taken exactly and therefore there could be more than one way to construct a minimum-edge-cut set. In this paper, we propose a new method to get a minimum-edge-cut set by selecting edges of a graph in...

1. Let G = (V,E) be a c-edge connected graph. In other words, assume that the size of minimum cut in G is at least c. Construct a graph G′(V,E ′) by sampling each edge of G with probability p independently at random and reweighing each edge with weight 1/p. Suppose c > log n, and is such that 10 log(n) c 2 ≤ 1. Show that as long as p ≥ 10 log(n) c 2 , with high probability the size of every cut...

For a graph G, the P2-path graph, P2(G), has for vertices the set of all paths of length 2 in G. Two vertices are connected when their union is a path or a cycle of length 3. We present lower bounds on the edge-connectivity, (P2(G)) of a connected graph G and give conditions for maximum connectivity. A maximally edge-connected graph is superif each minimum edge cut is trivial, and it is optimum...

In the last few years, computer vision and image processing techniques have been developed to solve many problems. One of them, graph cut method is powerful optimization technique for minimizing energy function. And as you know, many edge detectors are already advanced. Edge detector is widely used in computer vision to find object boundaries in images. But traditional edge detectors detect all...

Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G−F has no isolated vertex. The restricted edge connectivity λ′ is theminimumcardinality over all restricted edge cuts.WecallG λ′-optimal if λ′ = ξ , where ξ is theminimum edgedegree inG. Moreover, a λ′-optimal graphG is called ...

The response of ground-dwelling spider assemblages to edges created by wildfire was compared to their response to clear-cut edges in black spruce forests in eastern Canada. For each disturbance six edge transects 100 m long were established. Spiders were collected with pitfall traps 10 m apart from 50 m inside the disturbances to 50 m in the forest interior. Measurements of the forest floor str...

Given a graph with non-negative edge weights, the MAXCUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in different subsets is maximized. This classical NP-hard problem finds applications in VLSI design, statistical physics, and classification among other fields. This paper compares the performance of several greedy construct...