Published in:
J. Combinatorial Theory, Ser. B., Vol. 100, 1-22, 2010.
A two-dimensional
mixed framework is a pair (G,p), where G=(V;D,L) is a graph whose
edges are labeled as `direction' or `length' edges, and
p is a map from V to $R^{2}.
The label of an edge uv represents a direction or length constraint
between p(u) and p(v). The framework (G,p) is called
globally rigid if every framework (G,q) in which
the direction or length between the
endvertices of
corresponding
edges is the same as in (G,p),
can be obtained from (G,p) by
a translation and, possibly, a dilation by -1.
We characterize the generically globally rigid mixed
frameworks (G,p) for which the edge set of
G is a circuit in the associated direction-length rigidity matroid.
We show that such a framework is globally rigid if and only
if each 2-separation S of G is `direction balanced', i.e.
each `side' of S contains a direction edge.
Our result is based on a new inductive construction for
the family of edge-labeled graphs which satisfy these hypotheses.
We also settle a related open problem posed by
Servatius and Whiteley concerning the inductive
construction of circuits in the direction-length rigidity matroid.
Bibtex entry:
@techreport{egres-08-09,
AUTHOR | = | {Jackson, Bill and Jord{\'a}n, Tibor}, |
TITLE | = | {Globally Rigid Circuits of the Direction-Length Rigidity Matroid}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2008}, |
NUMBER | = | {TR-2008-09} |