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) 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 ...

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