A minimal cut set is a minimal set of reactions whose inactivation would guarantee a failure in a certain network function or functions. Minimal cut sets (MCSs) were initially developed from the metabolic pathway analysis method (MPA) of elementary modes (EMs); they provide a way of identifying target genes for eliminating a certain objective function from a holistic perspective that takes into...

We study the minimal enumeration degree (e-degree) problem in models of fragments of Peano arithmetic (PA) and prove the following results: In any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal e-degree. By contrast, this phenomenon fails in the absence of Σ2 induction. In fact, whether eve...

1 Minimum Cuts In this lecture we will describe an algorithm that computes the minimum cut (or simply mincut) in an undirected graph. A c u t is deened as follows. by S is the subset of edges (ii j) 2 E such that jfii jg \ Sj = 1. That is, (S) consists of all those edges with exactly one endpoint in S. Given an undirected graph G = (V E) and for each edge e 2 E a nonnegative cost (or capacity) ...

The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity was not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first...

In 2014, some scholars showed that every 2-connected claw-free graph G with independence number α(G)≤3 is Hamiltonian one exception of family graphs. If a nontrivial path contains only internal vertices degree two and end not two, then we call it branch. A set S branches called branch cut if delete all edges leading to more components than G. We use bond denote minimal cut. branch-bond has an o...

The overset, or chimera, grid methodology utilizes a set of overlapping grids to discretize the solution domain. The critical first step in the overset process is the hole cutting with many different approaches currently in use. The present paper investigates an approach that seeks to maximize automation and minimize user inputs. The method uses the surfaces of the grids that define geometry fo...

We show that the following two problems are polynomiaUy equivalent: 1) Given a (weighted) graph G, and a cut C of G, decide whether C is maximal or not . 2) Given a (weighted) graph G, and a cut C of G, decide whether C is maximal or not , and in case it is not , find a better solution C'. As a consequence, an optimali ty testing oracle may be used to design a polynomial t ime algori thm for ap...

We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem: on query S ⊂ V , the oracle returns the size of the cut between S and V \ S. We provide algorithms computing an exact minimum s-t cut in G with Õ(n5/3) queries, and computing an exact global minimum cut ...