The source location problem is a problem of computing a minimum cost source set in an undirected graph so that the connectivity between the source set and a vertex is at least the demand of the vertex. In this paper, the connectivity between a source set S and a vertex v is defined as the maximum number of paths between v and S no two of which have common vertex except v. We propose an O(d∗ log...

Finding a combinatorial test for rigidity in 3D is an open problem. We prove that vertex connectivity cannot be used to construct such a test by describing a class of mechanisms that increase the vertex connectivity of flexible graphs to 5. Our result is tight, as minimally rigid graphs in 3D can be at most 5-connected.

In the vertex connectivity survivable network design problem we are given an undirected graph G = (V,E) and connectivity requirement r(u, v) for each pair of vertices u, v ∈ V .We are also given a cost function on the set of edges. Our goal is to find the minimum cost subset of edges such that for every pair (u, v) of vertices we have r(u, v) vertex disjoint paths in the graph induced by the ch...

We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc” wireless networks. The graphs are obtained as “irrigation subgraphs” of the well-known random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors” of each vertex and the number of edges c that each vertex is allowed to send to these....

We consider the variant of the minimum vertex cover problem in which we require that the cover induces a connected subgraph. We give new approximation results for this problem in dense graphs, in which either the minimum or the average degree is linear. In particular, we prove tight parameterized upper bounds on the approximation returned by Savage’s algorithm, and extend a vertex cover algorit...

A graph G is almost 4-connected if it is simple, 3-connected, has at least five vertices, and V (G) cannot be partitioned into three sets A,B,C in such a way that |C| = 3, |A| ≥ 2, |B| ≥ 2, and no edge of G has one end in A and the other end in B. A graph K is a subdivision of a graph G if K is obtained from G by replacing its edges by internally disjoint nonzero length paths with the same ends...

The Randić index R(G) of a graph G is defined by R(G) = ∑ uv 1 √ d(u)d(v) , where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n ≥ 3 vertices with the Randić index R(G) and the diameter D(G), R(G) − D(G) ≥ √ 2 − n+1 2 and R(G...

Let G(n; m) be a connected graph without loops and multiple edges which has n vertices and m edges. We ÿnd the graphs on which the zeroth-order connectivity index, equal to the sum of degrees of vertices of G(n; m) raised to the power − 1 2 , attains maximum.

New lower bounds for eigenvalues of a simple graph are derived. Upper and lower bounds for eigenvalues of bipartite graphs are presented in terms of traces and degree of vertices. Finally a non-trivial lower bound for the algebraic connectivity of a connected graph is given.