A covering graph is a graph map that induces a bijection on the incident oriented edges at every vertex. Connected based covers of a graph correspond to subgroups of its fundamental group.
For and a generating set , the Schreier coset graph has vertices and a directed -edge from to for every . For a free group, this is the covering graph associated with .
The core of a based covering graph is the union of all reduced loops through its base vertex. A subgroup of a finitely generated free group is finitely generated exactly when its core graph is finite.
