Forbidden Graph Characterizations
Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski's theorem and Wagner's theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K4 or the complete bipartite graph K2,3. Alternatively, a graph is outerplanar if and only if it does not contain K4 or K2,3 as a minor, a graph obtained from it by deleting and contracting edges.
A triangle-free graph is outerplanar if and only if it does not contain a subdivision of K2,3.
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