Undirected Vertex-Connectivity Structure and Smallest Four-Vertex-Connectivity Augmentation
نویسنده
چکیده
In this paper we study properties for the structure of an undirected graph that is not vertex connected We also study the evolution of this structure when an edge is added to optimally increase the vertex connectivity of the underlying graph Several properties reported here can be extended to the case of a graph that is not k vertex connected for an arbitrary k Using properties obtained here we solve the problem of nding a smallest set of edges whose addition vertex connects an undirected graph This is a fundamental problem in graph theory and has applications in network reliability and in statistical data security We give an O n log n m time algorithm for nding a set of edges with the smallest cardinality whose addition vertex connects an undirected graph where n and m are the number of vertices and edges in the input graph respectively This is the rst polynomial time algorithm for this problem when the input graph is not vertex connected We also show a formula to compute this smallest number in O n n n m time where is the inverse of the Ackermann function This is also the rst polynomial time algorithm for computing this number when the input graph is not vertex connected Our algorithm can also be used to nd a smallest k vertex connectivity augmentation for any k
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