The Jordan-Brouwer theorem for graphs

We prove a discrete Jordan-Brouwer-Schoenflies separation theorem telling that a (d − 1)-sphere H embedded in a d-sphere G defines two different connected graphs A,B in G such a way that A ∩B = H and A ∪B = G and such that the complementary graphs A,B are both d-balls. The graph theoretic definitions are due to Evako: the unit sphere of a vertex x of a graph G = (V,E) is the graph generated by {y |; (x, y) ∈ E}. Inductively, a finite simple graph is called contractible if there is a vertex x such that both its unit sphere S(x) as well as the graph generated by V \ {x} are contractible. Inductively, still following Evako, a dsphere is a finite simple graph for which every unit sphere is a (d− 1)-sphere and such that removing a single vertex renders the graph contractible. A d-ball B is a contractible graph for which each unit sphere S(x) is either a (d− 1)-sphere in which case x is called an interior point, or S(x) is a (d − 1)-ball in which case x is called a boundary point and such that the set δB of boundary point vertices generates a (d − 1)-sphere. These inductive definitions are based on the assumption that the empty graph is the unique (−1)-sphere and that the one-point graph K1 is the unique 0-ball and that K1 is contractible. The theorem needs the following notion of embedding: a sphere H is embedded in a graph G if it is a subgraph of G and if any intersection with any finite set of mutually neighboring unit spheres is a sphere. A knot of codimension k in G is a (d − k)-sphere H embedded in a d-sphere