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- W2022859983 abstract "The problem of combinatorially determining the rank of the 3-dimensional bar-joint rigidity matroid of a graph is an important open problem in combinatorial rigidity theory. Maxwell's condition states that the edges of a graph G=(V,E) are independent in its d-dimensional generic rigidity matroid only if (a) the number of edges |E|⩽d|V|−(d+12), and (b) this holds for every induced subgraph with at least d vertices. We call such graphs Maxwell-independent in d dimensions.2 Laman's theorem shows that the converse holds for d=2 and thus every maximal Maxwell-independent set of G has size equal to the rank of the 2-dimensional generic rigidity matroid. While this is false for d=3, we show that every maximal, Maxwell-independent set of a graph G has size at least the rank of the 3-dimensional generic rigidity matroid of G. This answers a question posed by Tibór Jordán at the 2008 rigidity workshop at BIRS [1]. Along the way, we construct subgraphs (1) that yield alternative formulae for a rank upper bound for Maxwell-independent graphs and (2) that contain a maximal (true) independent set. We extend this bound to special classes of non-Maxwell-independent graphs. One further consequence is a simpler proof of correctness for existing algorithms that give rank bounds." @default.
- W2022859983 created "2016-06-24" @default.
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- W2022859983 date "2014-03-01" @default.
- W2022859983 modified "2023-09-26" @default.
- W2022859983 title "Maxwell-independence: a new rank estimate for the 3-dimensional generic rigidity matroid" @default.
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- W2022859983 doi "https://doi.org/10.1016/j.jctb.2013.12.001" @default.
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