Topological properties of the intersection graph of covers of n-dimensional surfaces

Motivated by a problem in computer graphic we develop discrete models of continuous n-dimensional spaces by using molecular spaces and graphs. We study a family of induced subgraphs of a given graph and find the conditions when the intersection graph of this family is homotopic to the given graph. We show that for a given surface and for all proper covers of this surface their intersection graphs are homotopic, can be transformed from one to the other by contractible transformations and have the same Euler characteristic and homology groups. As an example, we consider discrete two-dimensional closed spaces that are digital counterparts of a two-dimensional sphere, a torus, a projective plane and a Klein bottle.