Forbidden Induced Subgraphs for Perfect Matchings

Let F be a family of connected graphs. A graph G is said to be F-free if G is H-free for every graph H in F. We study the problem of characterizing the families of graphs F such that every large enough connected F-free graph of even order has a perfect matching. This problems was previously studied in Plummer and Saito (J Graph Theory 50(1):1-12, 2005), Fujita et al. (J Combin Theory Ser B 96(3):315-324, 2006) and Ota et al. (J Graph Theory, 67(3):250-259, 2011), where the authors were able to characterize such graph families F restricted to the cases {pipe}F{pipe} ≤ 1, {pipe}F{pipe} ≤ 2 and ≤ 3, respectively. In this paper, we complete the characterization of all the families that satisfy the above mentioned property. Additionally, we show the families that one gets when adding the condition {pipe}F{pipe} ≤ k for some k ≥ 4.