## Characterization of Uniquely Representable Graphs. (arXiv:1708.01272v1 [math.CO])

The betweenness structure of a finite metric space $M = (X, d)$ is a pair
$\mathcal{B}(M) = (X,\beta_M)$, where $\beta_M = \{(x, y, z)\in X^3 : d(x, z) =
d(x, y) + d(y, z)\}$ is the so-called betweenness relation of $M$. The
adjacency graph of a betweenness structure $\mathcal{B} = (X,\beta)$ is the
simple graph $G(\mathcal{B}) = (X, E)$ where the edges are such pairs of
distinct points for which no third point lies between them. A connected graph
is \emph{uniquely representable} if it is the adjacency graph of a unique
betweenness structure. It was known before that trees are uniquely
representable. In this paper, we give a full characterisation of uniquely
representable graphs by showing that they coincide with the so-called Husimi
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