The structure of neighbor disconnected vertex transitive graphs Original Research Article

We define a graph to be neighbor disconnected if the removal of the closed neighborhood of a vertex leaves a disconnected induced subgraph. Our main theorem is that a vertex transitive graph is neighbor disconnected if and only if it is a wreath product of vertex transitive graphs, with the necessary restriction that one factor must be neighbor disconnected whenever the other factor is a clique. Among the applications, we describe all connected neighbor disconnected vertex transitive graphs of degree not exceeding 10, and characterize the generating sets of all neighbor disconnected Cayley graphs.