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.
Is the sequence $f(m,n)$ f(n,m)$ unimodal for fixed $n$? Can one find asymptotics for values such as $f\left(n,\frac{n(n-1)}{4}\right)$ when $n$ is large?
