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Graphs that do not contain a cycle with a node that has at least two neighbors on it.
Aboulker P., Radovanovic M., Trotignon N., Vuskovic K.
Siam Journal on Discrete Mathematics 26, 4 (2012) 1510-1531 [ensl-00800037 - version 1]
fulltext access On wheel-free graphs
Aboulker P., Havet F., Trotignon N.
N° RR-7651 (2011) [inria-00602079 - version 1]