We consider the graphs Hn a defined as the Cartesian products of n complete graphs with a vertices each. Let an edge cut partition the vertex set of a graph into k subsets A1, . . . , Ak with ||Ai| − |Aj || ≤ 1. We consider the problem of determining the minimal size of such a cut for the graphs defined above and present bounds and asymptotic results for some specific values of k.

Let an edge cut partition the vertex set of a graph into k disjoint subsets A1, . . . , Ak with ||Ai| − |Aj || ≤ 1. We consider the problem of determining the minimal size of such a cut for a given graph. For this we introduce a new lower bound method which is based on the solution of an extremal set problem and present bounds for some graph classes based on Hamming graphs.

The laser cutting parameters are dependent on the beam characteristics, the cutting rate required, the composition and thickness of the material to be cut, and the desired cut edge quality. The laser cutting process and cut quality depend upon the proper selection of laser and workpiece parameters. Deficiencies in cutting quality may be related to the slow process drifts and disturbances that a...

Let G be a graph. A component of G is a maximal connected subgraph in G. A vertex v is a cut vertex of G if κ(G− v) > κ(G), where κ(G) is the number of components in G. Similarly, an edge e is a bridge of G if κ(G− e) > κ(G). In this paper, we will propose new O(n) algorithms for finding cut vertices and bridges of a trapezoid graph, assuming the trapezoid diagram is given. Our algorithms can b...

The 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 ...

A graph G is said to be hyper-connected if the removal of every minimum cut creates exactly two connected components, one of which is an isolated vertex. In this paper, we first generalize the concept of hyper-connected graphs to that of semi-hyper-connected graphs: a graph G is called semi-hyper-connected if the removal of every minimum cut of G creates exactly two components. Then we characte...

Given an edge-weighted graph and a subset of k vertices called terminals, a multiway cut is a partition of the vertices into k components, each containing exactly one terminal. The multiway cut problem is to find a multiway cut minimizing the sum of the weights of edges with endpoints in different components. Recently, Călinescu et al. described an approximation algorithm based on a geometric e...

For an edge weighted undirected graph G and an integer k ≥ 2, a k-way cut is a set of edges whose removal leaves G with at least k components. We propose a simple approximation algorithm to the minimum k-way cut problem. It computes a nearly optimal k-way cut by using a set of minimum 3-way cuts. We show that the performance ratio of our algorithm is 2−3/k for an odd k and 2− (3k−4)/(k2−k) for ...

Given a graph G on a vertex set V , MaxCut is the problem of finding a partition of V so that the number of edges cut is as large as possible. MaxCut has a simple random approximation algorithm. Given a graph G = (V,E), output a random partition (R,B) of V . Specifically, for each vertex v ∈ V , put v in R or B with probability 1 2 . Theorem 3. The expected number of edges cut by this algorithm...

In this note we solve the edge-connectivity augmentation problem over symmetric parity families. It provides a solution for the minimum T-cut augmentation problem. We also extend a recent result of C. Q. Zhang [8].