  We introduce a new model construction for Martin-L\"{o}f intensional type
theory, which is sound and complete for the 1-truncated version of the theory.
The model formally combines the syntactic model with a notion of realizability;
it also encompasses the well-known Hofmann- Streicher groupoid semantics. As
our main application, we use the model to analyse the syntactic groupoid
associated to the type theory generated by a graph G, showing that it has the
same homotopy type as the free groupoid generated by G.
