Conceptually graph vulnerability relates to the study of graphintactness when some of its elements are removed. The motivation forstudying vulnerability measures is derived from design and analysisof networks under hostile environment. Graph tenacity has been anactive area of research since the the concept was introduced in1992. The tenacity T(G) of a graph G is defined asbegin{center} $T(G)=di...

The super edge-connectivity λ′ of a connected graph G is the minimum cardinality of an edge-cut F in G such that every component of G − F contains at least two vertices. LetGi be a connected graph with order ni , minimum degree δi and edge-connectivity λi for i = 1, 2. This article shows that λ′(G1 × G2) ≥ min{n1 λ2,n2 λ1,λ1 + 2λ2, 2λ1+λ2} forn1,n2 ≥ 3 andλ′(K2×G2) = min{n2, 2λ2}, which general...

ÐGiven a graph G ˆ …V ;E† with n vertices and m edges, the k-connectivity of G denotes either the k-edge connectivity or the k-vertex connectivity of G. In this paper, we deal with the fully dynamic maintenance of k-connectivity of G in the parallel setting for k ˆ 2; 3. We study the problem of maintaining k-edge/vertex connected components of a graph undergoing repeatedly dynamic updates, such...

Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λr(G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this pap...

Given a simple graph G = (V,E), our goal is to find a smallest set F of new edges such that G = (V,E ∪ F ) is k-edge-connected and simple. Recently this problem was shown to be NP-complete. In this paper we prove that if OPT P is high enough—depending on k only—then OPT S = OPT P holds, where OPT S (OPT P ) is the size of an optimal solution of the augmentation problem with (without) the simpli...

In this note we provide a generalization of a result of Goddard, Raines and Slater [4] on edge-connectivity of permutation graphs for hypergraphs. A permutation hypergraph Gπ is obtained from a hypergraph G by taking two disjoint copies of G and by adding a perfect matching between them. The main tool in the proof of the graph result was the theorem on partition constrained splitting off preser...

The main subject of the thesis is connectivity augmentation: we would like to make a given graphk-connected by adding a minimum number of new edges. There are four basic problems in thisfield, since one might consider both edgeand node-connectivity augmentation in both graphsand digraphs. The thesis wishes to contribute to three out of these four problems: directed-and undirecte...

Ota and Tokuda [2] gave a minimum degree condition for a K1,n-free graph to have a 2factor. Though their condition is best-possible, their sharpness examples have edge-connectivity one. In this paper, we improve their minimum degree condition for K1,n-free graphs with large connectivity or large edge-connectivity. Our bound is sharp, and together with Ota and Tokuda’s theorem, we completely det...