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We give a complete characterization of on-line arbitrarily vertex decomposable graphs in the family of unicycle graphs called suns. A sun is a graph with maximum degree three, such that deleting vertices of degree one results in a cycle. This result has already been used in another paper to prove some Ore-type conditions for on-line arbitrarily decomposable graphs.
A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n1,…,nk) of positive integers with n1+⋯+nk=n, there exists a partition (V1,…,Vk) of the vertex set of G such that Vi induces a connected subgraph of order ni, 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...
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