Geodesic
The Ryjáček Closure and a Forbidden Subgraph
Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 621-628

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The Ryjáček closure is a powerful tool in the study of Hamiltonian properties of claw-free graphs. Because of its usefulness, we may hope to use it in the classes of graphs defined by another forbidden subgraph. In this note, we give a negative answer to this hope, and show that the claw is the only forbidden subgraph that produces non-trivial results on Hamiltonicity by the use of the Ryjáček closure.
Keywords: closure, claw-free graph, Hamiltonian graph, perfect matching, traceable graph 
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Saito, Akira; Xiong, Liming. The Ryjáček Closure and a Forbidden Subgraph. Discussiones Mathematicae. Graph Theory, Tome 36 (2016) no. 3, pp. 621-628. http://geodesic.mathdoc.fr/item/DMGT_2016_36_3_a7/