Arbitrarily vertex decomposable suns with few rays

A graph GG of order nn is called arbitrarily vertex decomposable if for each sequence (n1,…,nk)(n1,…,nk) of positive integers with n1+⋯+nk=nn1+⋯+nk=n, there exists a partition (V1,…,Vk)(V1,…,Vk) of the vertex set of GG such that ViVi induces a connected subgraph of order nini, for all i=1,…,ki=1,…,k. A sun with rr rays is a unicyclic graph obtained by adding rr hanging edges to rr distinct vertices of a cycle. We characterize all arbitrarily vertex decomposable suns with at most three rays. We also provide a list of all on-line arbitrarily vertex decomposable suns with any number of rays.