Forbidden induced subgraphs for near perfect matchings

Let FF be a family of connected graphs. A graph GG is said to be FF-free if GG is HH-free for every graph HH in FF. Fujita et al. (2006) [2] studied the problem of characterizing the families of graphs FF such that every large enough connected FF-free graph of odd order has a near perfect matching. In the same work, the authors characterized such families FF, but in the case where every graph in FF is triangle-free. In this paper, we complete the characterization of all such families, removing the triangle-free condition.