Robbins' theorem

[1] An extension of Robbins' theorem to mixed graphs by Boesch & Tindell (1980) shows that, if G is a graph in which some edges may be directed and others undirected, and G contains a path respecting the edge orientations from every vertex to every other vertex, then any undirected edge of G that is not a bridge may be made directed without changing the connectivity of G. In particular, a bridgeless undirected graph may be made into a strongly connected directed graph by a greedy algorithm that directs edges one at a time while preserving the existence of paths between every pair of vertices; it is impossible for such an algorithm to get stuck in a situation in which no additional orientation decisions can be made.

A strong orientation of a given bridgeless undirected graph may be found in linear time by performing a depth-first search of the graph, orienting all edges in the depth-first search tree away from the tree root, and orienting all the remaining edges (which must necessarily connect an ancestor and a descendant in the depth-first search tree) from the descendant to the ancestor.

[3] Parallel algorithms are also known for finding strongly connected orientations of mixed graphs.

[4] Robbins originally motivated his work by an application to the design of one-way streets in cities.

This theory concerns the problem of making a square grid, constructed from rigid rods attached at flexible joints, rigid by adding more rods or wires as cross bracing on the diagonals of the grid.