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, ..., vn) of vertices of G such that
- vi and vi+1 are adjacent in G for all i, i.e., w is a walk
- vi-1 ≠ vi+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, ..., vn) is adjacent to the vertex (r, v1, v2, ..., vn-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, ..., vn) in T to the vertex vn in G.
Read more about this topic: Covering Graph
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