Acyclicity of Switching Classes

Abstract

For a finite undirected graphG= (V,E) and a subsetA⊆V, the vertex switching ofGbyAis defined as the graphGA= (V,E'), 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.