Distanced graphs

This paper studies model theoretic conditions that arise in the study of distanced graphs. These are graphs with additional relation symbols for finite distances. Every embedding of distanced graphs is an isometric embedding. The conditions we study are the version of homogeneity, injectivity and universality for distanced graphs. There is a countable graph which isometrically embeds every countable graph. There is a unique countable distace homogeneous graph with this universal property. Its first-order theory is the model completion of the theory of distanced graphs. It has a unique countable connected model, but it has more than one connected model of each uncountable cardinality.