  For a vertex set $S\subseteq V(G)$ in a graph $G$, the {\em distance
multiset}, $D(S)$, is the multiset of pairwise distances between vertices of
$S$ in $G$. Two vertex sets are called {\em homometric} if their distance
multisets are identical. For a graph $G$, the largest integer $h$, such that
there are two disjoint homometric sets of order $h$ in $G$, is denoted by
$h(G)$. We slightly improve the general bound on this parameter introduced by
Albertson, Pach and Young (2010) and investigate it in more detail for trees
and graphs of bounded diameter. In particular, we show that for any tree $T$ on
$n$ vertices $h(T) \geq \sqrt[3]{n}$ and for any graph $G$ of fixed diameter
$d$, $h(G) \geq cn^{1/ (2d-2)}$.
