Showing papers on "Connectivity published in 1973"

Coxeter functors and gabriel's theorem

Abstract: It has recently become clear that a whole range of problems of linear algebra can be formulated in a uniform way, and in this common formulation there arise general effective methods of investigating such problems. It is interesting that these methods turn out to be connected with such ideas as the Coxeter—Weyl group and the Dynkin diagrams. We explain these connections by means of a very simple problem. We assume no preliminary knowledge. We do not touch on the connections between these questions and the theory of group representations or the theory of infinite—dimensional Lie algebras. For this see [3]—[5]. Let Γ be a finite connected graph; we denote the set of its vertices by Γο and the set of its edges by ΓΊ (we do not exclude the cases where two vertices are joined by several edges or there are loops joining a vertex to itself). We fix a certain orientation Λ of the graph Γ; this means that for each edge / e Γι we distinguish a starting-point a(/) e Γο and an end-point

An Algorithm for Determining Whether the Connectivity of a Graph is at Least k

Abstract: The algorithm presented in this paper is for testing whether the connectivity of a large graph of $n$ vertices is at least $k$. First the case of undirected graphs is discussed, and then it is shown that a variation of this algorithm works for directed graphs. The number of steps the algorithm requires, in case $k > \sqrt{n}$, is bounded by $O(kn^{3})$.

Heawood's theorem and connectivity

Abstract: Let G be a finite connected graph with no loops or multiple edges. The point-connectivity K ( G ) of G is the minimum number of points whose removal results in a disconnected or trivial graph. Similarly, the line-connectivity λ( G ) of G is the minimum number of lines whose removal results in a disconnected or trivial graph. For the complete graph K p we have

Graphs with given connectivity and independence number or networks with given measures of vulnerability and survivability

Abstract: The criterion for invulnerability of a network based on the connectivity of a graph is well treated in literature. We define a criterion for survivability of a network in terms of the independence number of a graph. The following problems are then considered. 1) Find an r -connected graph, with n vertices and m_0 = [(nr + l)/2] edges, whose independence number is the minimum possible, where [x] denotes the greatest integer less than or equal to x \cdot 2 ) Given positive integers n,m,r , and k , find the realizability conditions for a graph with n vertices and m edges, whose connectivity is at least r and independence number is at most k .