On intersection of three embedded spheres in 3-space

We study intersection of two polyhedral spheres without self-intersections in 3-space R 3 . We find necessary and sufficient conditions on sequences x → = ( x 1 , x 2 , … , x n ) , y → = ( y 1 , y 2 , … , y n ) of positive integers, for existence of polyhedral spheres f , g ⊂ R 3 such that• has n connected components, which can be numbered so that the i-th component has x i neighbors in f and has n connected components, which can be numbered so that the i-th component has y i neighbors in g. gously we study intersection of three polyhedral spheres without self-intersections in 3-space.