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Line 19: ==Applications== Robbins originally motivated his work by an application to the design of one-way streets in cities. Another application arises in [[structural rigidity]], in the theory of [[grid bracing]]. This theory concerns the problem of making a square ~~grid—constructed~~grid, constructed from rigid rods attached at flexible ~~joints—rigid~~joints, rigid by adding more rods or wires as [[cross bracing]] on the diagonals of the grid. A set of added rods makes the grid rigid if an associated undirected graph is connected, and is doubly braced (remaining rigid if any edge is removed) if in addition it is bridgeless. Analogously, a set of added wires (which can bend to reduce the distance between the points they connect, but cannot expand) makes the grid rigid if an associated directed graph is strongly connected.{{sfnp\|Baglivo\|Graver\|1983}} Therefore, reinterpreting Robbins' theorem for this application, the doubly braced structures are exactly the structures whose rods can be replaced by wires while remaining rigid. ==Notes==

Robbins' theorem: Difference between revisions