A k-tree is either a complete graph on (k+1) vertices or given a k-tree G' with n vertices, a k-tree G with (n+1) vertices can be constructed by introducing a new vertex v and picking a k-clique Q in G' and then joining each vertex u in Q. A graph G is k-edge-connected if the graph remains connected even after deleting fewer edges than k from the graph. A k-edge-connected graph G is said to be minimally k-connected if G \ {e} is no longer k-edge-connected for any edge e belongs to E(G) where E(G) denotes the set of edges of G. In this paper we find two separate O (n2) algorithms so that a minimally 2-connected graph can be obtained from a 2-tree and a minimally k-connected graph can be obtained from a k-tree. In a k-tree (k \geq 2) we find the edges which are insensitive to the k-connectivity have both their end vertices of degrees greater than or equal to k+1.This property is fully exploited to find an algorithm which reduces any k-tree to a minimally k-connected graph.

Note on minimally k-connected graphs

Self-assembly is a process in which simple objects autonomously combine themselves into larger objects. It is considered as a promising technique in nano-technology. Two simple graphs G1 and G2 with a clique of same size overlap and a new self-assembled graph is formed. Besides studying the properties of self assembled graphs on cliques, we answer the question: Can a given set of graphs be generated through the self-assembly of cliques? If so, how to find the generator that could generate the given set of graphs by the process of clique-self-assembly. The question of the existence of minimal generator is also discussed. The necessary and sufficient condition for a graph H to be obtained by the iterated clique-self-assembly of the graph G is also answered. We also conclude that the problem of finding the generator is decidable. We note the importance of our work with respect to several closely related clique finding problem.

Suresh Badarla

Papers

Note on minimally k-connected graphs

Clique-Detection Algorithm Using Clique-Self-Assembly