Adding into the comments by Alain Valette: ... Hence my suggestion of Strong Shape Theory. Vietoris homotopy gives one way of approaching strong shape. Again it is mentioned on the nLab. Vietoris homology still has the problem of non-exact sequences. It can be replaced by Steenrod-Sitnikov homology or Mardesic's strong homology theory. (There is also a renewal of interest in finite topological spaces, see work by Minian and Barmak. An application of similar ideas occurs through topological data analysis, see work by Carlsson et al at Stamford. This also uses a Rips complex, that is probably known to you from your general interests.)
Looking at some of the other comments and answers, it may help to look at some of the ideas of graph homotopy theory that are around. These are linked via the original work of Dowker (1953) on the homology of a relation. (I can provide more indicators if you think it would help.)
(Edit: The following describes a related theory: Perspectives on A-homotopy theory and its applications, Hélène BarceloaBarcelo, Reinhard Laubenbacher, Discrete Mathematics 298 (2005) 39 – 61.)
