Disconnecting two vertices s and z in a graph by removing minimum number of is fundamental problem algorithmic theory. This (s, z)-Separation well-known to be polynomial solvable serves as an important primitive many applications related network connectivity.

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. Our deterministic structure processes a batch of d ≤ d? failed vertices in Õ(d) time and thereafter answers connectivity queries in O(d) time. It occupies space O(d?m log n). We develop a randomized Monte Carlo version of our data structure with update time Õ(d), query time O(d), an...

A subset T ⊆ V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v ∈ T ; T is k-connected in J if T is k-connected to every s ∈ T . We consider the Subset k-Connectivity Augmentation problem: given a graph G = (V,E) with edge/node-costs, node subset T ⊆ V , and a subgraph J = (V,EJ ) of G such that T is k-connected in J , ...

O s c i l l a t i o n s a n d s p i k e s t a t i s t i c s i n b i o p h y s i c a l a t t r a c t o r n e t w o r k s Abstract The work of this thesis concerns how cortical memories are stored and retrieved. In particular, large-scale simulations are used to investigate the extent to which associative attractor theory is compliant with known physiology and in vivo dynamics. The first question...

Let G = (V,E) be a connected graph. G is said to be super edge connected (or super-k for short) if every minimum edge cut of G isolates one of the vertex of G. A graph G is called m-super-k if for any edge set S # E(G) with jSj 6m, G S is still super-k. The maximum cardinality of m-super-k is called the edge fault tolerance of super edge connectivity of G. In this paper, we discuss the edge fau...

For each finite field F of prime order there is a constant c such that every 4-connected matroid has at most c inequivalent representations over F. We had hoped that this would extend to all finite fields, however, it was not to be. The (m,n)-mace is the matroid obtained by adding a point freely to M(Km,n). For all n ≥ 3, the (3, n)-mace is 4-connected and has at least 2 representations over an...

We show that every 4-connected planar graph has a B3-EPG representation, i.e., every vertex is represented by a curve on the grid with at most three bends, and two vertices are adjacent if and only if the corresponding curves share an edge of the grid. Our construction is based on a modification of the representation by touching thickened L-shapes proposed by Gonçalves et al. [4].