Arbitrarily vertex decomposable suns with few rays

A graph G of order n is called arbitrarily vertex decomposable if for each sequence ( n 1 , … , n k ) of positive integers with n 1 + ⋯ + n k = n , there exists a partition ( V 1 , … , 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.