T-irreducible extensions for starlike trees

We deal with a sort of optimal extensions of graphs, so called T-irreducible extensions. T-irreducible extension of a graph $G$ is an extension of $G$ obtained by removing a maximal set of edges from the trivial extension of $G$. A difficult starlike tree is a starlike tree that has at least one difficult node. T-irreducible extensions for nondifficult starlike trees were constructed by M. B. Abrosimov, T-irreducible extensions for palms (one of subclasses of starlike trees) were constructed by S. G. Kurnosova. Counterexamples were found to a method of Harary and Khurum, who tried to construct possible T-irreducible extensions for starlike trees. T-irreducible extensions for difficult starlike trees are constructed.