The connectivity of \(Z\)-transformation graphs of perfect matchings of polyominoes (Q1815325)

Let \(G\) be a plane graph. The \(Z\)-transformation graph \(Z(G)\) of \(G\) is the graph whose vertices are the perfect matchings of \(G\) with two vertices being adjacent if and only if the symmetric difference of the corresponding perfect matchings consists of the boundary of a single finite face of \(G\). The author discusses some properties of polyominoes with perfect matchings. In particular, he proves that outside of two exceptional cases the connectivity of the \(Z\)-transformation graph of a polyomino with perfect matchings has connectivity equal to its minimum degree.