I will call two graphs $G$ and $H$, $r$-equidecomposable (in analogy with Hilbert's third problem) if they can be written as unions of disjoint subgraphs $$G\cong \bigsqcup_{i=1}^r G_i\quad ,\quad H\cong \bigsqcup_{i=1}^r H_i$$ (disjoint here means they have no common edges), and $G_i\cong H_i$ for $1\le i \le r$. Let $\epsilon(G,H)$ be the smallest $r$ for which $G$ and $H$ are $r$-equidecomposable. Let $$f(n,m)=\max_{G,H\in V(n,m) } \epsilon(G,H)$$ where $V(n,m)$ is the collection of all graphs with $n$ vertices and $m$ edges.