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A cycle is a polygon tree. A new polygon tree $G′$ can be created out of an existing polygon tree $G$ by adding a cycle which shares exactly one edge with graph $G$.

I want to know which graph class it belongs to.

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There are many ways to describe graphs, so asking which "graph class" these graphs belong to is ambiguous, since it is included in many: the class of $2$-connected graphs that are not $3$-connected, for example, or the class of graphs that are at most $3$-colorable. – Andrew Salmon Aug 16 '13 at 0:05
Thanks Andrew, actually I want to know which standard graph classes it includes and/or a subset of – sgpl Aug 16 '13 at 4:09
It seems the following. By induction we can easily prove that all polygon trees are planar. When all added cycles are triangles, then we obtain exactly 2-trees with at least 3 vertices (see more about 2-trees, for instance, at p.10 of our paper). – Alex Ravsky Aug 17 '13 at 9:27

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