Abstract
We investigate algorithmic questions and structural problems concerning graph families defined by ‘edge-counts’. Motivated by recent developments in the unique realization problem of graphs, we give an efficient algorithm to compute the rigid, redundantly rigid, M-connected, and globally rigid components of a graph. Our algorithm is based on (and also extends and simplifies) the idea of Hendrickson and Jacobs, as it uses orientations as the main algorithmic tool.
We also consider families of bipartite graphs which occur in parallel drawings and scene analysis. We verify a conjecture of Whiteley by showing that 2d-connected bipartite graphs are d-tight. We give a new algorithm for finding a maximal d-sharp subgraph. We also answer a question of Imai and show that finding a maximum size d-sharp subgraph is NP-hard.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berg, A., Jordán, T.: A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid. J. Combinatorial Theory, Ser. B. 88, 77–97 (2003)
Connelly, R.: On generic global rigidity, Applied geometry and discrete mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 147–155. Amer. Math. Soc., Providence (1991)
Frank, A., Gyárfás, A.: How to orient the edges of a graph, Combinatorics (Keszthely). Coll. Math. Soc. J. Bolyai 18, 353–364 (1976)
Gabow, H.N., Westermann, H.H.: Forests, frames and games: Algorithms for matroid sums and applications. Algorithmica 7, 465–497 (1992)
Graver, J., Servatius, B., Servatius, H.: Combinatorial Rigidity. AMS Graduate Studies in Mathematics, vol. 2 (1993)
Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21(1), 65–84 (1992)
Hendrickson, B., Jacobs, D.: An algorithm for two-dimensional rigidity percolation: the pebble game. J. Computational Physics 137, 346–365 (1997)
Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Comput. 2, 135–158 (1973)
Imai, H.: On combinatorial structures of line drawings of polyhedra. Discrete Appl. Math. 10, 79 (1985)
Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs, EGRES Technical Report 2002-12, submitted to J. Combin. Theory Ser. B, www.cs.elte.hu/egres/
Jacobs, D.J., Thorpe, M.F.: Generic rigdity percolation: the pebble game. Phys. Rev. Lett. 75, 4051 (1995)
Laman, G.: On graphs and rigidity of plane skeletal structures. J. Engineering Math. 4, 331–340 (1970)
Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebraic Discrete Methods 3(1), 91–98 (1982)
Sugihara, K.: On some problems in the design of plane skeletal structures. SIAM J. Algebraic Discrete Methods 4(3), 355–362 (1983)
Sugihara, K.: Machine interpretation of line drawings. MIT Press, Cambridge (1986)
Whiteley, W.: Parallel redrawing of configurations in 3-space, preprint, Department of Mathematics and Statistics, York University, North York, Ontario (1987)
Whiteley, W.: A matroid on hypergraphs with applications in scene analysis and geometry. Discrete Comput. Geometry 4, 75–95 (1989)
Whiteley, W.: Some matroids from discrete applied geometry. Matroid theory, Seattle, WA, 1995, pp. 171–311, Contemp. Math., 197, Amer. Math. Soc., Providence (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berg, A.R., Jordán, T. (2003). Algorithms for Graph Rigidity and Scene Analysis. In: Di Battista, G., Zwick, U. (eds) Algorithms - ESA 2003. ESA 2003. Lecture Notes in Computer Science, vol 2832. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39658-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-39658-1_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20064-2
Online ISBN: 978-3-540-39658-1
eBook Packages: Springer Book Archive