Robbins' theorem: Difference between revisions

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==Algorithms and complexity==

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.<ref>{{harvtxt|Vishkin|1985}} credits this observation to {{harvtxt|Atallah|1984}}, and {{harvtxt|Balakrishnan|1996}} credits it to {{harvtxt|Roberts|1978}}. But as {{harvtxt|Clark|Holton|1991}} point out, the same algorithm is already included (with the assumption of [[k-vertex-connected graph|2-vertex-connectivity]] rather than 2-edge-connectivity) in the seminal earlier work of {{harvtxt|Hopcroft|Tarjan|1973}} on depth -first search.</ref> Although this algorithm is not suitable for [[parallel computer]]s, due to the difficulty of performing depth -first search on them, alternative algorithms are available that solve the problem efficiently in the parallel model.<ref>{{harvtxt|Vishkin|1985}}.</ref> Parallel algorithms are also known for finding strongly connected orientations of mixed graphs.<ref>{{harvtxt|Soroker|1988}}.</ref>

==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 from rigid rods attached at flexible 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==

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| title = Parallel strong orientation of an undirected graph

| volume = 18

| year = 1984| url = https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1361&context=cstech }}.

| year = 1984}}.

*{{citation

| last1 = Baglivo | first1 = Jenny A. | author1-link = Jenny Baglivo

| last2 = Graver | first2 = Jack E.

| contribution = 3.10 Bracing structures

| isbn = 9780521297844

| pages = 76–87

| publisher = Cambridge University Press | ___location = Cambridge, UK

| series = Cambridge Urban and Architectural Studies

| title = Incidence and Symmetry in Design and Architecture

| title-link = Incidence and Symmetry in Design and Architecture

| year = 19841983}}.

*{{citation

| last = Balakrishnan | first = V. K.

Line 91 ⟶ 105:

| title = Algorithm 447: efficient algorithms for graph manipulation

| volume = 16

| year = 1973}}.| s2cid = 14772567 | doi-access = free

}}.

*{{citation

| last = Robbins | first = H. E. | author-link = Herbert Robbins

Line 100 ⟶ 115:

| volume = 46

| issue = 5 | year = 1939

| doi=10.2307/2303897| hdl = 10338}}.dmlcz/101517

| hdl-access = free

}}.

*{{citation

| last = Roberts | first = Fred S. | authorlink = Fred S. Roberts