Covering Graph - Universal Cover

Universal Cover

For any connected graph G, it is possible to construct its universal covering graph. This is an instance of the more general universal cover concept from topology; the topological requirement that a universal cover be simply connected translates in graph-theoretic terms to a requirement that it be acyclic and connected; that is, a tree. The universal covering graph is unique (up to isomorphism). If G is a tree, then G itself is the universal covering graph of G. For any other finite connected graph G, the universal covering graph of G is a countably infinite (but locally finite) tree.

The universal covering graph T of a connected graph G can be constructed as follows. Choose an arbitrary vertex r of G as a starting point. Each vertex of T is a non-backtracking walk that begins from r, that is, a sequence w = (r, v₁, v₂, ..., v_n) of vertices of G such that

• v_i and v_i+1 are adjacent in G for all i, i.e., w is a walk
• v_i-1 ≠ v_i+1 for all i, i.e., w is non-backtracking.

Then, two vertices of T are adjacent if one is a simple extension of another: the vertex (r, v₁, v₂, ..., v_n) is adjacent to the vertex (r, v₁, v₂, ..., v_n-1). Up to isomorphism, the same tree T is constructed regardless of the choice of the starting point r.

The covering map f maps the vertex (r) in T to the vertex r in G, and a vertex (r, v₁, v₂, ..., v_n) in T to the vertex v_n in G.