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 $H\in H$. 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 $\delta \left(G\right)\ge 2$ 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 $\delta \left(G\right)\ge 2$ and every odd branch-bond of G has an edge branch.