Towards an optimal algorithm for recognizing Laman graphs
Vol. 13, no. 2, pp. 219-232, 2009. Concise paper.
Abstract A graph G with n vertices and m edges is a generically minimally rigid graph (Laman graph), if m=2n−3 and every induced subset of k vertices spans at most 2k−3 edges. Laman graphs play a fundamental role in rigidity theory.
We discuss the Verification problem: Given a graph G with n vertices, decide if it is Laman. We present an algorithm that recognizes Laman graphs in O(Tst(n)+n logn) time, where Tst(n) is the best time to extract two edge disjoint spanning trees from a graph with n vertices and 2n−2 edges, or decide no such trees exist. So far, it is known that Tst(n) is O(n3/2√{logn}).
Submitted: April 2008.
Reviewed: December 2008.
Revised: January 2009.
Reviewed: May 2009.
Revised: May 2009.
Accepted: July 2009.
Final: July 2009.
Published: July 2009.
Communicated by Ioannis G. Tollis
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