On minimal vertex 1-extensions of special type graph union

In 2001, it was conjectured that the minimal vertex 1-extension of a graph $G+G^*$, where $G^*$ is a minimal vertex 1-extension of graph $G$, is unique up to isomorphism and has the form $G^*+G^*$. We construct two counterexamples to this conjecture showing that, in general, it is wrong. Also, we show that the statement is true for many graphs.