A Characterization of Uniquely Representable Graphs

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.Comment: 16 pages (without references); 3 figures; major changes: simplified proofs, improved notations and namings, short overview of metric graph theor

Bell inequality and common causal explanation in algebraic quantum field theory

Bell inequalities, understood as constraints between classical conditional probabilities, can be derived from a set of assumptions representing a common causal explanation of classical correlations. A similar derivation, however, is not known for Bell inequalities in algebraic quantum field theories establishing constraints for the expectation of specific linear combinations of projections in a quantum state. In the paper we address the question as to whether a 'common causal justification' of these non-classical Bell inequalities is possible. We will show that although the classical notion of common causal explanation can readily be generalized for the non-classical case, the Bell inequalities used in quantum theories cannot be derived from these non-classical common causes. Just the opposite is true: for a set of correlations there can be given a non-classical common causal explanation even if they violate the Bell inequalities. This shows that the range of common causal explanations in the non-classical case is wider than that restricted by the Bell inequalities