where the minimum is taken over all subsets X of E(G) such that ω(G − X) − c > 0. In this paper, we establish a relationship 7 between λc(G) and τc−1(G), which gives a characterization of the edge-connectivity of a graph G in terms of the spanning tree 8 packing number of subgraphs of G. The digraph analogue is also obtained. The main results are applied to show that if a graph G is 9 s-hamilto...

The main subject of the thesis is connectivity augmentation: we would like to make a given graph k-connected by adding a minimum number of new edges. There are four basic problems in this field, since one might consider both edgeand node-connectivity augmentation in both graphs and digraphs. The thesis wishes to contribute to three out of these four problems: directedand undirected node-connect...

Let G be a graph and T a set of vertices. A T -path in G is a path that begins and ends in T , and none of its internal vertices are contained in T . We define a T -path covering to be a union of vertex-disjoint T paths spanning all of T . Concentrating on graphs that are tough (the removal of any nonempty set X of vertices yields at most |X| components), we completely characterize the edges th...

The generalized k-connectivity κk(G) of a graph G was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized k-edge-connectivity, λk(G). For general k, the generalized k-edgeconnectivity of a complete graph is obtained. For k ≥ 3, tight upper and lower bounds of κk(G) and λk(G) are given for a connected graph G of order n, namely, 1 ≤ κk(G) ≤ n− k2 and 1 ≤...

For a connected graph G = (V, E), an edge set S ⊆ E is a k-restricted edge cut if G−S is disconnected and every component of G− S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G) = min{|[X, X ]| : |X | = k,G[X ] is connected}. G is λk -optimal if λk(G) = ξk(G). Moreover, G is super-λk ...

The objective of the thesis is to discuss edge-connectivity and related connectivity conceptsin the context of undirected and directed hypergraphs. In particular, we focus on k-edge-connectivity and (k, l)-partition-connectivity of hypergraphs, and (k, l)-edge-connectivityof directed hypergraphs. A strong emphasis is placed on the role of submodularity in thestructural aspects o...

This paper considers the concept of restricted edge-connectivity, and relates that to the edgedegree of a connected graph. The author gives some necessary conditions for a graph whose restricted edge-connectivity is smaller than its minimum edge-degree, then uses these conditions to show some large classes of graphs, such as all connected edge-transitive graphs except a star, and all connected ...

We consider connectivity-augmentation problems in a setting where each potential new edge has a nonnegative cost associated with it, and the task is to achieve a certain connectivity target with at most p new edges of minimum total cost. The main result is that the minimum cost augmentation of edge-connectivity from k − 1 to k with at most p new edges is fixed-parameter tractable parameterized ...

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are the classical results of Tutte and Nash-Williams from 1961 which show that a λ...

In the edge connectivity augmentation problem one wants to find an edge set of minimum total capacity that increases the edge connectivity of a given undirected graph by τ . It is a known non-trivial property of the edge connectivity augmentation problem that there is a sequence of edge sets E1, E2, . . . , such that ⋃ i≤τ Ei optmially increases the connectivity by τ , for any integer τ . The m...