On arbitrarily 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 ${\displaystyle {\sum }^k\_\{i=1\}{n}_i=n}$, there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ {1,...,k} the set ${\displaystyle V_i}$ induces a connected subgraph of G on ${\displaystyle n_i}$ 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.