The solvability problem of a linear active network is approached from a purely topological point of view using the two-graph method. It can be said that a topological condition for the solvability is the existence of a common tree of the voltage and current graphs. A few conditions for the existence of a common tree are derived. If there exists no common tree, subgraphs which cause the nonexistence can be distinguished, and a partition of two-graphs can be introduced. The partition has similar properties to the principal partition of a graph or the canonical form of a bipartite graph, and a structure of two-graphs represented by a partial ordering of sets of edges can be defined. An algorithm to find the partition and a common tree, if one exists, or if no common tree exists, a tree of one of the graphs which has as many common edges as possible with a tree of the other graph, is given. The decomposition of the coefficient matrix accompanying the structure is discussed, and algorithms to determine the decomposition is given.
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