, Volume 25, Issue 2, pp 139144
First online:
On Characterizations of Rigid Graphs in the Plane Using Spanning Trees
 Sergey BeregAffiliated withDepartment of Computer Science, University of Texas at Dallas
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We study characterizations of generic rigid graphs and generic circuits in the plane using only few decompositions into spanning trees. Generic rigid graphs in the plane can be characterized by spanning tree decompositions [5,6]. A graph G with n vertices and 2n − 3 edges is generic rigid in the plane if and only if doubling any edge results in a graph which is the union of two spanning trees. This requires 2n − 3 decompositions into spanning trees. We show that n − 2 decompositions suffice: only edges of G − T can be doubled where T is a spanning tree of G.
A recent result on tensegrity frameworks by Recski [7] implies a characterization of generic circuits in the plane. A graph G with n vertices and 2n − 2 edges is a generic circuit in the plane if and only if replacing any edge of G by any (possibly new) edge results in a graph which is the union of two spanning trees. This requires \((2n2)({n\choose 2}  1)+1\) decompositions into spanning trees. We show that 2n − 2 decompositions suffice. Let \(e_1,e_2,\ldots,e_{2n2}\) be any circular order of edges of G (i.e. \(e_0 = e_{2n2}\)). The graph G is a generic circuit in the plane if and only if \(G + e_i  e_{i1}\) is the union of two spanning trees for any \(i = 1,2, \ldots, 2n2\). Furthermore, we show that only n decompositions into spanning trees suffice.
Keywords
Rigid graphs Circuits Tensegrity frameworks Title
 On Characterizations of Rigid Graphs in the Plane Using Spanning Trees
 Journal

Graphs and Combinatorics
Volume 25, Issue 2 , pp 139144
 Cover Date
 200905
 DOI
 10.1007/s0037300808362
 Print ISSN
 09110119
 Online ISSN
 14355914
 Publisher
 Springer Japan
 Additional Links
 Topics
 Keywords

 Rigid graphs
 Circuits
 Tensegrity frameworks
 Industry Sectors
 Authors

 Sergey Bereg ^{(1)}
 Author Affiliations

 1. Department of Computer Science, University of Texas at Dallas, Box 830688, Richardson, TX, 75083, USA