Certifying and constructing minimally rigid graphs in the plane

We study minimally rigid graphs in the plane or plane isostatic graphs. These graphs (also called Laman graphs) admit characterizations based on decomposition into trees (Crapo's theorem and Récski's theorem). Tree partitions can be viewed as certificates of plane isostatic graphs. Unfortunately, they require Ω(n2) time to verify their validity where n is the number of vertices in the graph. We present a new construction (which can be viewed as a hierarchical decomposition of the graph) called red-black hierarchy that (i) is a certificate for plane isostatic graphs, and (ii) can be verified in linear time. We also show that it can be computed in O(n2) time.A classical result in Rigidity Theory by Henneberg [9] states that the plane isostatic graphs can be constructed incrementally by special vertex insertions. We study the following computational problem: given a Laman graph G, compute a sequence of Henneberg insertions that yields G. We show that the red-bl ack hierarchy can be used to compute a Henneberg construction in O(n2) time. Applied to planar graphs our algorithm can speed up a recent algorithm by Haas et al. [8] for embedding a planar Laman graph as a pointed pseudo-triangulation by a factor of O(n).

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