Induced cycle structure and outerplanarity

Those nonseparable graphs whose induced cycles form a (necessarily minimum length) cycle basis are characterized in several ways — each a generalization of outerplanar graphs. For instance, they are the series–parallel graphs that do not contain a subdivision of K2,3 as an induced subgraph — whereas the outerplanar graphs are known to be the series–parallel graphs that do not contain a subdivision of K2,3 as a subgraph. The approach uses a certain ‘tree structure’ such that the outerplanar graphs are those for which that tree structure is unique.