Forbidden pairs for the matching extendability of graphs with connectivity at least 2 or 3

Let H be a set of connected graphs. A graph G is said to be H-free if G does not contain any member of H as an induced subgraph. A graph G having a perfect matching is called k-extendable if every matching of size k in G can be extended to a perfect matching. In this paper, we characterize the sets H with |H|=2 such that every (k+1)-connected H-free graph G of sufficiently large order permitting a perfect matching is k-extendable for k∈{1,2}.