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acyclic adjacent algorithm augmenting path back edge bipartite graph circuit of G circuit subspace cocircuit coloring computing connected graph Consider construct contains Corollary corresponding defined denote directed circuits directed graph directed path directed s-t path dual edge set edge-disjoint edge-induced subgraph edges incident edges of G electrical network elements end vertices equal Eulerian graph exists following theorem fundamental circuit fundamental cutset graph G graph of Fig graph theory Hence in-degree incidence matrix independent set induced subgraph labeling layered network Lemma length Let G matroid maximal maximum flow maximum matching n-vertex nonzero number of edges number of vertices paths in G planar graph problem Proof prove push result ring sum saturated self-loops shortest paths shown in Fig simple graph spanning tree strongly connected strongly connected component subgraph of G submatrix subset Suppose transitive closure transport network tree of G undirected vector space vertex set vertices of G weight spanning tree
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