  The betweenness structure of a finite metric space $M = (X, d)$ is a pair
$\mathcal{B}(M) = (X,\beta_M)$ where $\beta_M$ is the so-called betweenness
relation of $M$ that consists of point triplets $(x, y, z)$ such that $d(x, z)
= d(x, y) + d(y, z)$. The underlying graph of a betweenness structure
$\mathcal{B} = (X,\beta)$ is the simple graph $G(\mathcal{B}) = (X, E)$ where
the edges are pairs of distinct points with no third point between them. A
connected graph $G$ is uniquely representable if there exists a unique metric
betweenness structure with underlying graph $G$. It was implied by previous
works that trees are uniquely representable. In this paper, we give a
characterization of uniquely representable graphs by showing that they are
exactly the block graphs. Further, we prove that two related classes of graphs
coincide with the class of block graphs and the class of distance-hereditary
graphs, respectively. We show that our results hold not only for metric but
also for almost-metric betweenness structures.
