Upper bound for the number of additional edges in minimal $1$-edge extensions of starlike trees

Minimal edge extension of graphs can be regarded as a model of optimal edge fault tolerant implementation of a system. This paper is about an upper bound for the number of additional edges in minimal $1$-edge extensions for graphs of a special class – starlike trees. Two schemes for constructing $1$-edge extensions for any kind starlike trees and an algorithm based on these schemes are proposed.