Characterization and recognition of partial 3-trees
SIAM Journal on Algebraic and Discrete Methods
Forbidden minors characterization of partial 3-trees
Discrete Mathematics
Algorithms finding tree-decompositions of graphs
Journal of Algorithms
SIAM Review
Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization
SIAM Journal on Optimization
Global Continuation for Distance Geometry Problems
SIAM Journal on Optimization
Semidefinite programming for ad hoc wireless sensor network localization
Proceedings of the 3rd international symposium on Information processing in sensor networks
Low-dimensional embedding with extra information
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Graph Minors. XX. Wagner's conjecture
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
Real-time tracking for sensor networks via sdp and gradient method
Proceedings of the first ACM international workshop on Mobile entity localization and tracking in GPS-less environments
Universal rigidity: towards accurate and efficient localization of wireless networks
INFOCOM'10 Proceedings of the 29th conference on Information communications
Universal Rigidity and Edge Sparsification for Sensor Network Localization
SIAM Journal on Optimization
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
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Recently, Connelly and Sloughter [14] have introduced the notion of d-realizability of graphs and have, among other things, given a complete characterization of the class of 3-realizable graphs. However, their work has left open the question of finding an algorithm for realizing those graphs. In this paper, we resolve that question by showing that the semidefinite programming (SDP) approach of [11, 32] can be used for realizing 3-realizable graphs. Specifically, we use SDP duality theory to show that given a graph G and a set of lengths on its edges, the optimal dual multipliers of a certain SDP give rise to a proper equilibrium stress for some realization of G. Using this result and the techniques in [14, 31], we then obtain a polynomial time algorithm for (approximately) realizing 3-realizable graphs. Our results also establish a little-explored connection between SDP and tensegrity theories and allow us to derive some interesting properties of tensegrity frameworks.