We discuss several parametrizations of the space of circular planar electrical networks. With any circular planar network we associate a canonical minimal network with the same response matrix, called a “standard” network. The conductances of edges in a standard network can be computed as a biratio of Pfaffians constructed from the response matrix. The conductances serve as coordinates that are compatible with the cell structure of circular planar networks in the sense that one conductance degenerates to $0$ or $\infty$ when moving from a cell to a boundary cell. We also show how to test if a network with $n$ nodes is well-connected by checking that $\binom{n}{2}$ minors of the $n\times n$ response matrix are positive; Colin de Verdière had previously shown that it was sufficient to check the positivity of exponentially many minors. For standard networks with $m$ edges, positivity of the conductances can be tested by checking the positivity of $m+1$ Pfaffians.

The authors study the problem of tiling a simple polygon of surface n with rectangles of given types (tiles). They present a linear time algorithm for deciding if a polygon can be tiled with 1 * m and k * 1 tiles (and giving a tiling when it exists), and a quadratic algorithm for the same problem when the tile types are m * k and k * m.>

Article Free Access Share on How to take short cuts Authors: Claire Kenyon IHES, 35 route de Chartres, 91440 Bures-sur-Yvette, France IHES, 35 route de Chartres, 91440 Bures-sur-Yvette, FranceView Profile , Richard Kenyon View Profile Authors Info & Claims SCG '91: Proceedings of the seventh annual symposium on Computational geometryJune 1991 Pages 250–255https://doi.org/10.1145/109648.109676Online:01 June 1991Publication History 0citation181DownloadsMetricsTotal Citations0Total Downloads181Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF