Closure and forbidden pairs for 2-factors

Pairs of connected graphs X,YX,Y such that a graph GG is 2-connected and XYXY-free implies that GG is hamiltonian were characterized by Bedrossian. Using the closure concept for claw-free graphs, the first author simplified the characterization by showing that if considering the closure of GG, the list in the Bedrossian characterization can be reduced to one pair, namely, K1,3,N1,1,1K1,3,N1,1,1 (where Ki,jKi,j is the complete bipartite graph, and Ni,j,kNi,j,k is the graph obtained by identifying endvertices of three disjoint paths of lengths i,j,ki,j,k to the vertices of a triangle). Faudree et al. characterized pairs X,YX,Y such that GG is 2-connected and XYXY-free implies that GG has a 2-factor. Recently, the first author et al. strengthened the closure concept for claw-free graphs such that the closure of a graph has stronger properties while still preserving the (non)-existence of a 2-factor. In this paper we show that, using the 2-factor closure, the list of forbidden pairs for 2-factors can be reduced to two pairs, namely, K1,4,P4K1,4,P4 and K1,3,N1,1,3K1,3,N1,1,3.