Geodesic
On arbitrarily vertex decomposable unicyclic graphs with dominating cycle
Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 403-412

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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 n_i = n, there exists a partition (V₁,...,Vₖ) of vertex set of G such that for every i ∈ 1,...,k the set V_i induces a connected subgraph of G on 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.
Keywords: arbitrarily vertex decomposable graph, dominating cycle 
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Cichacz, Sylwia; Zioło, Irmina. On arbitrarily vertex decomposable unicyclic graphs with dominating cycle. Discussiones Mathematicae. Graph Theory, Tome 26 (2006) no. 3, pp. 403-412. http://geodesic.mathdoc.fr/item/DMGT_2006_26_3_a4/