An explicit formula for the number of tunnels in digital objects
  An important concept in digital geometry for computer imagery is that of tunnel. In this paper we obtain a formula for the number of tunnels as a function of the number of the object vertices, pixels, holes, connected components, and 2x2 grid squares. It can be used to test for tunnel-freedom a digital object, in particular a digital curve. 
Valentin Brimkov, Angelo Maimone, Giorgio Nordo