On arbitraly vertex decomposable unicyclic graphs with dominating cycle

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1, . . . , nk) of positive integers such that ∑k i=1 ni = n, there exists a partition (V1, . . . , Vk) of vertex set of G such that for every i ∈ {1, . . . , k} the set Vi induces a connected subgraph of G on ni vertices. We consider arbitrarily vertex decomposable unicyclic graphs with dominating cycle. We also characterize all such graphs with at most four hanging vertices such that exactly two of them have a common neighbour.