Some Theorems on Detachments Preserving Local-Edge-Connectivity

A detachment operation transforms a given graph by splitting some vertices into several their copies while maintaining the original set of edges, whose end vertices may change after the operation. A graph obtained by such an operation is called a detachment of the graph. Although necessary and sufficient conditions for a graph to have a k-edge-connected detachment are known, it is left open to derive a necessary and sufficient condition for a graph to admit a detachment which has a specified local-edge-connectivity. In this paper, we show our recent results on related problems such as the problem of finding an Eulerian detachment of a graph that maximizes the local-edge-connectivity, and the problem of splitting a vertex s into two vertices of degree 3 and d(s) −3 preserving the local-edge-connectivity.