Vertices In Connected Graph Research Articles

Minimum Cost Range Assignment for the Vertex Connectivity of Graphs

Minimum Cost Range Assignment for the Vertex Connectivity of Graphs

On 2-Approximation to the Vertex-Connectivity in Graphs

On 2-Approximation to the Vertex-Connectivity in Graphs

Partition conditions and vertex-connectivity of graphs

It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev 1, ...,v n ≠V(G) and for any sequence of positive integersk 1, ...,k n such thatk 1+...+k n =|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv 1, ...,v(n), and the class of the partition containingv i induces a connected subgraph consisting ofk i vertices, fori=1, 2, ...,n. Now fix the integersk 1, ...,k n . In this paper we study what can we say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv 1, ...,v n ≠V(G). We find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases we give sharp lower bounds for the vertex-connectivity ofG.

Whitney Connectivity of Matroids

A new definition of matroid connectivity is introduced and its properties are investigated in this paper. Vertex connectivity of graphs is expressed in an algebraic form and generalized to matroids. This generalized connectivity is called the Whitney connectivity of matroids. It is shown that the Whitney connectivity of the polygon matroid of a graph is the same as the vertex connectivity of the graph provided the graph is connected. Various properties of Whitney matroid connectivity and comparison with Tutte connectivity are also examined.