Transitivity of local complementation and switching on graphs Original Research Article

The operations complementation C, local complementation λx, and switching σx for the vertices x of a finite undirected graph are considered. The operation λx complements the subgraph induced by the neighbourhood of x in the given graph, and the switching σx changes the neighbourhood of x to its complement vertex set. It is proved that the compositions δx=λxC (for vertices x∈D) generate a transitive group on the graphs with vertex set D, that is, for any two graphs g and h on D, there exists a composition α of operations δx such that h=α(g). It is also shown that the compositions τx=λxσx (for x∈D) generate a transitive group on the graphs.