A series-reduced tree is defined as a tree with no vertices of degree 2. The word "reduced" suggests that there is a process for reducing trees that have vertices of degree 2. It seems to me that this process would be to replace any vertex chain with a single edge. For example, all path graphs with more than 2 vertices would be reduced to $P_2$. Is there a name for this process? If so, I can't seem to find it.
Also, I can't find anything about series-reduced graphs in general, even though the definition can be easily extended. One could define a series-reduced graph as a graph with no vertices of degree 2. Similarly, I can imagine a process for reducing graphs by replacing any vertex chain with a single edge. Although, I can see problems with extending this to graphs because I'm not sure how a cycle graph would be reduced. Would it reduce to the null graph? Is my extension to graphs in general valid, or are there problems with it I didn't anticipate?
Has this all been established and I just don't know the correct terminology?
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