Arbitrarily vertex decomposable suns with few rays

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,…,n_k) of positive integers with n₁+?+n_k=n, there exists a partition (V₁,…,V_k) of the vertex set of G such that V_i induces a connected subgraph of order n_i, for all i=1,…,k. A sun with r rays is a unicyclic graph obtained by adding r hanging edges to r 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.