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Planar electric networks. II. (Reseaux électriques planaires. II.) (French) Zbl 0853.05074
A planar electrical network is a planar graph \((V,E)\) having so-called boundary points \(V_0\) and an interior part \(V \backslash V_0\), moreover associated to the edges are the inverses of the resistence \(\rho (e)\) (also called the conductivity). To every boundary point in \(V_0\) there is associated an electrical potential due to which there are incoming and outgoing electrical flows at the boundary points. The authors introduce so called elementary transformations on the interior of electrical planar networks, that when applied can be used to simplify a network by decreasing the number of edges. The planar electrical networks now can be viewed as the nodes of a graph where two nodes (networks) are joined if one can be transformed into the other by an elementary electrical transformation. The connected components in this graph can be used to define an equivalence relation: two networks are equivalent if and only if they are in the same connected component.
The first theorem states that for every planar electrical network there exists a sequence of elementary transformations such that after each transformation the number of edges decreases until it reaches a minimum. Moreover two minimal networks in the same equivalence class can be transformed into each by applying a number of elementary transformations during which the number of edges remains constant. In the second theorem well connected planar networks are characterized, i.e. the minimal graphs in the equivalence class of a planar network having \(N\) boundary points have \(N(N - 1)/2\) edges if the network is well connected. As a consequence if the minimal graph in an equivalence class has less than \(N(N - 1)/2\) edges the graphs in this class are not well connected. The most important theorem of the paper is Theorem 4 which states that two planar electrical networks are in the same equivalence class if and only if they have the same number of boundary points and the same in and outgoing flows at these points given the electrical potentials at the boundary points. In the last section the authors describe the connection between rectangular tiles of a polygon in which a conductivity is associated to the tiles, the horizontal edges of the tiles are shortcuts and the vertical edges are isolators. The authors show how the elementary transformations can be described in this case.
The example given in Figure 16 is incorrect, namely the edge with number 7 should start at the node in between edges 4 and 5 and not at the node with edges 1, 4, 12 and 15.

MSC:
05C90 Applications of graph theory
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
31A25 Boundary value and inverse problems for harmonic functions in two dimensions
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