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 trees. 查看全文>>