# Strongly maximal H-free spanning subgraph

Let the graphs $G=(V,E)$ and $H$ be fixed. An edge set $F\subseteq E$ is called $H$-free if $(V,F)$ does not contain $H$ as a subgraph. We say that $F$ is strongly maximal if for any $H$-free edge set $I\subseteq E$ we have $\left|I \setminus F\right| \leq \left|F \setminus I\right|$.
If $H$ is the path of length two, then we talk about strongly maximal matchings, and any $G$ admits such a matching ([1] p. 16. Theorem 5.6). For what other graphs $H$ can we guarantee the existence of a strongly maximal $H$-free $F$ for any $G$?