The Seven Graphs Whose \(H\)-Transformations Are Uniquely Determined

An \(H\)-transformation on a simple \(3\)-connected cubic planar graph \(G\) is the dual operation of flip flop on the triangulation \(G^*\) of the plane, where \(G^*\) denotes the dual graph of \(G\). We determine the seven \(3\)-connected cubic planar graphs whose \(H\)-transformations are uniquely determined up to isomorphism.