Characterizing forbidden pairs for the existence of even factors

Let H be a class of given graphs. We say that a graph is H-free if it contains no induced subgraph isomorphic to H for every H∈H. In 2017, the second author characterized all connected graphs H with order at least three such that every H-free graph G has an even factor if and only if δ(G)≥2 and every odd branch-bond of G has an edge branch. In this paper, we consider the case H is disconnected and characterize all the pairs {R,S} of connected graphs such that every {R,S}-free graph G has an even factor if and only if δ(G)≥2 and every odd branch-bond of G has an edge branch.