Definition 1.0.1 Let G = (V, E) be a graph with vertex set V and edge set E.
(1) A circuit is a connected 2-regular graph.
(2) A graph (subgraph) is even if the degree of each vertex is even.
(3) A bridge (or cut-edge) of a graph G is an edge whose removal increases the number of components of G (equivalently, a bridge is an edge that is not contained in any circuit of G).
Note that, in much of the literature related to circuit covers and integer flows, an even graph/subgraph is also called a cycle, which is adapted from matroid ...