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.