combinatorial conditions for the rigidity of tensegrity frameworks andrás recski budapest...
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![Page 1: Combinatorial conditions for the rigidity of tensegrity frameworks András Recski Budapest University of Technology and Economics La Vacquerie-et-Saint-](https://reader035.vdocument.in/reader035/viewer/2022062313/56649d3f5503460f94a1954c/html5/thumbnails/1.jpg)
Combinatorial conditions for the rigidity of
tensegrity frameworks
András RecskiBudapest University of Technology and Economics
La Vacquerie-et-Saint-Martin-de-Castries,
Languedoc-Roussillon, 2007
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Bar and joint frameworks
• Deciding the rigidity of a framework (that is, of an actual realization of a graph) is a problem in linear algebra.
• Deciding whether a graph is generic rigid, is a problem in combinatorics (discrete mathematics).
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What can a combinatorialist do?
• (S)he either studies „very symmetric” structures, like square or cubic grids,
• or prefers „very asymmetric” ones, that is, the generic structures.
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Three parts of my talk:
• One − story buildings as special tensegrity frameworks
• Tensegrity frameworks on the line
• Some results on the plane (to be continued by T. Jordán and Z. Szabadka today afternoon)
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Tensegrity frameworks
Rigid rods are resistant to compressions and tensions:
║xi − xk║= cik
Cables are resistant to tensions only:
║xi − xk║≤ cik
Struts are resistant to compressions only:
║xi − xk║≥ cik
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Frameworks composed from rods (bars), cables and struts are called tensegrity frameworks.
A more restrictive concept is the r-tensegrity framework, where rods are not allowed, only cables and struts. (The letter „r” means rod − free or restricted.)
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Rigidity of square grids
• Bolker and Crapo, 1977: A set of diagonal bars makes a k X ℓ square grid rigid if and only if the corresponding edges form a connected subgraph in the bipartite graph model.
• Baglivo and Graver, 1983: In case of diagonal cables, strong connectedness is needed in the (directed) bipartite graph model.
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Minimum # diagonals needed:
B = k + ℓ − 1 diagonal bars
C = 2∙max(k, ℓ ) diagonal cables
(If k ≠ ℓ then C − B > 1)
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Rigidity of one-story buildings
Bolker and Crapo, 1977: If each external vertical wall contains a diagonal bar then instead of studying the roof of the building one may consider a k X ℓ square grid with its four corners pinned down.
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Rigidity of one-story buildings
Bolker and Crapo, 1977: A set of diagonal bars makes a k X ℓ square grid (with corners pinned down) rigid if and only if the corresponding edges in the bipartite graph model form either a connected subgraph or a 2-component asymmetric forest.
For example, if k = 8, ℓ = 18, k ' = 4, ℓ ' = 9, then the 2-component forest is symmetric (L = K, where ℓ ' / ℓ = L, k' / k = K).
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Minimum # diagonals needed:
B = k + ℓ − 2 diagonal bars
C = k + ℓ − 1 diagonal cables (except if k = ℓ = 1 or k = ℓ =2)
(Chakravarty, Holman, McGuinness and R., 1986)
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Rigidity of one-story buildings
Which (k + ℓ − 1)-element sets of cables make the k X ℓ square grid (with corners pinned down) rigid?
Let X, Y be the two colour classes of the directed bipartite graph. An XY-path is a directed path starting in X and ending in Y.
If X0 is a subset of X then let N(X0) denote the set of those points in Y which can be reached from X0 along XY-paths.
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R. and Schwärzler, 1992:
A (k + ℓ − 1)-element set of cables makes the k X ℓ square grid (with corners pinned down) rigid if and only if
|N(X0)| ∙ k > |X0| ∙ ℓ
holds for every proper subset X0 of X and
|N(Y0)| ∙ ℓ > |Y0| ∙ k
holds for every proper subset Y0 of Y.
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Which one-story building is rigid?
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Which one-story building is rigid?
5
5
12
13
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Solution:
Top: k = 7, ℓ = 17, k0 = 5, ℓ0 = 12, L < K (0.7059 < 0.7143)
Bottom: k = 7, ℓ = 17, k0 = 5, ℓ0 = 13, L > K (0.7647 > 0.7143)
where ℓ0 / ℓ = L, k0 / k = K .
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Hall, 1935 (König, 1931):
A bipartitebipartite graph with colour classes X, Y has a perfect matching if and only if
|N(X0)| ≥ |X0|
holds for every proper subset X0 of X andand
|N(Y0)| ≥ |Y0|
holds for every proper subset Y0 of Y.
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Hetyei, 1964:
A bipartite graph with colour classes X, Y has perfect matchings and every edge is contained in at least one if and only if
|N(X0)| > |X0|
holds for every proper subset X0 of X andand
|N(Y0)| > |Y0|
holds for every proper subset Y0 of Y.
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From now on, generic structures
(well, most of the time…)
A one-dimensional bar-and-joint framework is rigid if and only if its graph is connected.
In particular, a graph corresponds to a one-dimensional minimally rigid bar-and-joint framework if and only if it is a tree.
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R. – Shai, 2005:
Let the cable-edges be red, the strut-edges be blue (and replace rods by a pair of parallel red and blue edges).
The graph with the given tripartition is realizable as a rigid tensegrity framework in the 1-dimensional space if and only if
• it is 2-edge-connected and
• every 2-vertex-connected component contains edges of both colours.
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The necessity of the conditions is trivial.
For the sufficiency recall the so called
ear-decomposition of 2-vertex-connected
graphs:
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Realization of a single circuit
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Adding ears (and stars)
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Towards the 2-dimensional case
• Shortly recall the characterizations of the graphs of rigid bar-and-joint frameworks.
• Special case: minimal generic rigid graphs (when the deletion of any edge destroys rigidity).
• Consider at first the simple trusses:
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Simple trusses (in the plane)
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All the simple trusses satisfy e = 2n – 3 (n ≥ 3) and they are minimally rigid,
but not every minimally rigid framework is a simple truss (consider the Kuratowski graph K3,3, for example).
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A famous minimally rigid structure:
Szabadság Bridge, Budapest
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Does e = 2n-3 imply that the framework is minimally rigid?
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Certainly not:
If a part of the framework is „overbraced”, there will be a nonrigid part somewhere else…
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Maxwell (1864):
If a graph G is minimal generic rigid in the plane then,
in addition to e = 2n – 3, the relation e' ≤ 2n' – 3 must
hold for every (induced) sub-graph G' of G.
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Laman (1970):
A graph G is minimal generic rigid in the plane if and only if
e = 2n – 3 and the relation e' ≤ 2n' – 3 holds for every (induced) subgraph G' of G.
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This is a „good characterization” of minimal generic rigid graphs in the plane, but we do not wish to check some 2n subgraphs…
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Lovász and Yemini (1982):
A graph G is minimal generic rigid in the plane if and only if
e = 2n – 3 and doubling any edge the resulting graph, with 2∙(n − 1) edges, is the union of two edge-disjoint trees.
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A slight modification (R.,1984):
A graph G is minimal generic rigid in the plane if and only if
e = 2n – 3 and, joining any two vertices with a new edge, the resulting graph, with 2∙(n − 1) edges, is the union of two edge-disjoint trees.
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Maxwell (1864):
If a graph G is minimal generic rigid in the space then,
in addition to e = 3n – 6, the relation e' ≤ 3n' – 6 must
hold for every (induced) sub-graph G' of G.
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However, the 3-D analogue of Laman’s theorem is not true:
The double banana graph (Asimow – Roth, 1978)
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We wish to generalize the above results for tensegrity frameworks:
When is a graph minimal ge-neric rigid in the plane as a tensegrity framework (or as an r-tensegrity framework)?
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Which is the more difficult problem?
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Which is the more difficult problem?
If rods are permitted then why should one use anything else?
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Which is the more difficult problem?
If rods are permitted then why should one use anything else?
„Weak” problem: When is a graph minimal generic rigid in the plane as an r-tensegrity framework?
„Strong” problem: When is a graph with a given tripartition minimal generic rigid in the plane as a tensegrity framework?
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An example to the
2-dimensional case:
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The graph K4 can be realized as a rigid tensegrity framework with struts {1,2}, {2,3} and {3,1} and with cables for the rest (or vice versa) if ‘4’ is in the convex hull of {1,2,3} …
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…or with cables for two independent edges and struts for the rest (or vice versa) if none of the joints is in the convex hull of the other three.
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As a more difficult example, consider the emblem of the Sixth Czech-Slovak Inter-national Symposium on Com-binatorics, Graph Theory, Algorithms and Applications
(Prague, 2006, celebrating the 60th birthday of Jarik Nešetřil).
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If every bar must be replaced by a cable or by a strut then only one so-lution (and its reversal) is possible.
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Critical rods cannot be replaced by cables or struts if we wish to preserve rigidity
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Jordán – R. – Szabadka, 2007
A graph can be realized as a rigid d-dimensional r-tensegrity framework
if and only if it can be realized as a rigid d-
dimensional rod framework and none of its edges are critical.
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Corollary (Laman – type):
A graph G is minimal generic rigid in the plane as an r-tensegrity framework if and only if
e = 2n – 2 and the relation e' ≤ 2n' – 3 holds for every proper subgraph G' of G.
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Corollary (Lovász-Yemini – type):
A graph is minimal generic rigid in the plane as an r-tensegrity framework if and only if it is the union of two edge-disjoint trees and remains so if any one of its edges is moved to any other position.
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The „strong problem” is open for the 2-dimensional case.
Here is a more complicated
example.
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Rigidity and connectivity
• A graph is generic rigid in the 1-dimensional space as an r-tensegrity framework if and only if it is 2-edge-connected.
• For the generic rigidity in the plane as an r-tensegrity framework, a graph must be 2-vertex-connected and 3-edge-connected. However, neither 3-vertex-connectivity nor 4-edge-connectivity is necessary.
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A rigid r-tensegrity framework. Its graph is neither 3-vertex-
connected, nor 4-edge-connected.
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To be continued
in the afternoon…