Closure and Forbidden Pairs for Hamiltonicity

Let C be the claw K1,3 and N the net, i.e. the only connected graph with degree sequence 333111. It is known (Bedrossian, Thesis, Memphis State University, USA, 1991; Faudree and Gould, Discrete Math.173 (1997), 45–60) that if X,Y is a pair of connected graphs, then, for any 2-connected graph G, G being XY-free implies G is hamiltonian if and only if X is the claw C and Y belongs to a finite list of graphs, one of them being the net N. For any such pair X,Y we show that the closures of all 2-connected XY-free graphs form a subclass of the class of CN-free graphs, and we fully describe their structure.