Most terminology and notation in this book follow standard textbooks in graph theory (such as [18], [19], [41], [242], etc.). Some terminology is slightly different from those classical textbooks.

Definition 1.0.1 Let G = (V, E) be a graph with vertex set V and edge set E.

1. (1) A circuit is a connected 2-regular graph.

2. (2) A graph (subgraph) is even if the degree of each vertex is even.

3. (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 theory, and is different from what is used in many other graph theory textbooks. For the sake of less confusion, we will use even graph/subgraph instead of cycle in this book.

Definition 1.0.2 (1) A family ℱ of circuits (or even subgraphs) of a graph G is called a circuit cover (or an even subgraph cover) if every edge of G is contained in some member(s) of ℱ.

(2) A circuit cover (or an even subgraph cover) ℱ of a graph G is a double cover of G if every edge is contained in precisely two members of ℱ.