  Discrete homotopy theory or A-homotopy theory is a combinatorial homotopy
theory defined on graphs, simplicial complexes and metric spaces, reflecting
information about their connectivity. The purpose of the present paper is to
further understanding of the similarities between the A-homotopy and ordinary
homotopy theories. More precisely, we define the notion of a mapping fiber in
the category of graphs and study its basic properties, providing
non-compatibility with projection.
