We consider aspects of vertex splitting and connectivity augmentation in hypergraphs. In the splitting problem we are given a hypergraph G = (V + s, E) in which s is only incident with edges of size two and asked to “split” s in such a way that we preserve the local-connectivity in the set V . We develop results that lead to a characterisation of the hypergraphs in which there is no such split....

We study the problem of computing a minimum cut in a simple, undirected graph and give a deterministic O(m log n log log n) time algorithm. This improves both on the best previously known deterministic running time of O(m log n) (Kawarabayashi and Thorup [12]) and the best previously known randomized running time of O(m log n) (Karger [11]) for this problem, though Karger’s algorithm can be fur...

A graph is locally connected if for each vertex v of degree --2, the subgraph induced by the vertices adjacent to v is connected . In this paper we establish a sharp threshold function for local connectivity . Specifically, if the probability of an edge of a labeled graph of order n is p = ((3/2 + s„) log n/n)'/ 2 where s„ _ ( log log n + log(3/8) + 2x)/(2 log n), then the limiting probability ...

Connectivity is an important measurement for the fault tolerance in interconnection networks. It is known that the augmented cube AQn is maximally connected, i.e. (2n − 1)-connected, for n ≥ 4. By the classical Menger’s Theorem, every pair of vertices in AQn is connected by 2n − 1 vertex-disjoint paths for n ≥ 4. A routing with parallel paths can speed up transfers of large amounts of data and ...

For a graph G, a detachment operation at a vertex transforms the graph into a new graph by splitting the vertex into several vertices in such a way that the original graph can be obtained by contracting all the split vertices into a single vertex. A graph obtained from a given graph G by applying detachment operations at several vertices is called a detachment of graph G. We consider a detachme...

Let G = (V + s,E) be a graph and let S = (d1, ..., dp) be a set of positive integers with ∑ dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with d(sj) = dj , 1 ≤ j ≤ p. Given a requirement function r(u, v) on pairs of vertices of V , an S-detachment is called r-admissible if the detached graph G satisfies λG′(x, y) ≥ r(x, y) for every pair x, y ∈ V . Here λH...

The study of ground spaces of local Hamiltonians is a fundamental task in condensed matter physics. In terms of computational complexity theory, a common focus has been to estimate a given Hamiltonian’s ground state energy. However, from a physics perspective, it is often more relevant to understand the structure of the ground space itself. In this paper, we pursue this latter direction by intr...

Models of associative memory usually have full connectivity or if diluted, random symmetric connectivity. In contrast, biological neural systems have predominantly local, non-symmetric connectivity. Here we investigate sparse networks of threshold units, trained with the perceptron learning rule. The units are given position and are arranged in a ring. The connectivity graph varies between bein...

Department of Biological and Medical Psychology, University of Bergen, Norway Default-mode network (DMN) functional connectivity and its task-dependent downregulation have attracted a lot of attention in the field of neuroscience. Nevertheless, the exact underlying mechanisms of DMN functional connectivity, or more specifically, the blood oxygen level-dependent (BOLD) signal, are still not comp...