Acyclicity of Switching Classes

For a finite undirected graphG = (V,E) and a subsetA ⊆ V, the vertex switching ofGbyAis defined as the graphGA = (V,Eʹ), which is obtained fromGby removing all edges betweenAand its complementAand adding as edges all nonedges betweenAandA. The switching class [G] determined byGconsists of all vertex switchingsGAfor subsetsA ⊆ V. We prove that the trees of a switching class [G] are isomorphic to each other. We also determine the types of treesTthat have isomorphic copies in [G]. Finally we show that apart from one exceptional type of forest, the real forests in a switching class are isomorphic. Here a forest is real, if it is disconnected.