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We recall several known results about minimally 2-connected graphs, and showthat they all follow from a decomposition theorem. Starting from an analogywith critically 2-connected graphs, we give structural characterizations of theclasses of graphs that do not contain as a subgraph and as an induced subgraph,a cycle with a node that has at least two neighbors on the cycle. From thesecharacterizations we get polynomial time recognition algorithms for theseclasses, as well as polynomial time algorithms for vertex-coloring andedge-coloring.

Graphs that do not contain a cycle with a node that has at least two neighbors on it