(The) graph augmentation problem in regular graphs

Abstract

The connectivity problem has been studied for a long time in the field of graph theory. Especially edge augmentation problems have been actively studied because; public organizations can supplement the defects of network nodes by solving them. Since the edge-connectivity of graph denotes edge disjoint paths between two vertex sets in G, adding supplement edges can reduce the congestions of the network which topology is identical to G. Also, it can be applied to physical and virtual network design. This thesis proposes the regular graph edge augmentation problem and the existence of a polynomial time algorithm which solves this problem. The regular graph edge augmentation problem, unlike other edge augmentation problems which insert minimum edges to meet target edge-connectivity, requires $\frag {|V (G)|} {2}$ more edges and makes the graph G maximum connected in $O(dn^5)$ time. We introduce the Edge Exchange algorithm. This algorithm generates a maximum connected (d + 1)-regular graph or a regular graph with the number of its minimum edge cut reduced.