On graphs and rigidity of plane skeletal structures

By G. Laman. 1970

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Citations in the corpus

An Overview of Recent Progress in the Study of Distributed Multi-Agent Coordination (2012, background): “According to [189], a graph ofn agents is rigid if at least 2n− 3 edge distances are available.”
On the collapse of locally isostatic networks (2009, methodology): “The Maxwell rule was generalized by Laman’s theorem (Laman 1970), which solves the problem of generic rigidity in 2D (rigidity that depends only on the topology of the system but does not depends on the specific edge lengths nor joint coordinates).”
Location, Localization, and Localizability (2010, background): “A graph is called generically rigid if one cannot continuously deform any of its realizations in the plane while preserving distance constraints([70,72]).”
Pseudo-Triangulations - a Survey (2006, methodology): “n abstract graph is minimally rigid if it is rigid but the removal of any edge invalidates this property. The fundamental theorem of rigidity theory in the plane can now be stated: Theorem 6.1 (Laman [35]). A graph is minimally rigid iﬀ it is a Laman graph. 6.2. Pointed Pseudo-Triangulation Graphs a”
Generic combinatorial rigidity of periodic frameworks (2010, methodology): “Whiteley developed the subject [26, Section 4] and its generalizations to higher-dimensional scene analysis [26, Section 8], and the idea of deducing the Maxwell–Laman Theorem from a direct proof of the Parallel Redrawing Theorem appears in [27].”
Notes on the Rigidity of Graphs (2007, background): “Theorem 2.3 implies that all independent graphs are Laman.”
Control of Minimally Persistent Formations in the Plane (2009, background): “This conclusion is a standard result of the rigid graph theory and is due to [19, 18].”
Rigidity, computation, and randomization in network localization (2004, methodology): “Rigidity has been long studied in mathematics and structural engineering (see for example [14], [10], [44], [9], [11]) and has a surprising number of applications in many areas.”
Geometric representations of graphs (2014, background): “Generic rigidity in the plane Laman [111] gave a characterization of generically stress-free graphs in the plane.”
Directed graphs for the analysis of rigidity and persistence in autonomous agent systems (2007, background): “A graph obtained by adding one vertex to a graph G = (V,E) and two edges from this vertex to other vertices is rigid if and only if G is rigid [1].”
Minimally rigid periodic graphs (2011, background): “In spite of a long history of attempts, the only satisfactory answer known at this time is Laman’s theorem for planar linkages [7].”
Combinatorial Lefschetz theorems beyond positivity (2018, background): “n as Laman’s rigidity criterion for planar graphs. Recall that a Laman graph is a graph G on nvertices and 2n−3 edges such that every subgraph on xvertices has at most 2x−3 edges. Theorem 4.1 (Laman, [Lam70]). Consider a Laman graph G. Then a generic realization of G in R2 with straight edges is inﬁ”
A Theory of Network Localization (2006, methodology): “Rigidity has been long studied in mathematics and structural engineering (see, for example, [40], [60], [31], [51], [61]) and has a surprising number of applications in many areas.”
Rigid graph control architectures for autonomous formations (2008, background): “A key result is Laman’s theorem [20], for which no three-dimensional analog exists.”