Geodesic
Extremal problems for forbidden pairs that imply hamiltonicity
Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 13-29

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Let C denote the claw K_1,3, N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and Z_i the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free (does not contain an induced copy of C or of Y) will be determined for Y a connected subgraph of either P₆, N, W, or Z₃. It should be noted that the pairs of graphs CY are precisely those forbidden pairs that imply that any 2-connected graph of order at least 10 is hamiltonian. These extremal numbers give one measure of the relative strengths of the forbidden subgraph conditions that imply a graph is hamiltonian. 
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Faudree, Ralph; Gyárfás, András. Extremal problems for forbidden pairs that imply hamiltonicity. Discussiones Mathematicae. Graph Theory, Tome 19 (1999) no. 1, pp. 13-29. http://geodesic.mathdoc.fr/item/DMGT_1999_19_1_a1/