A multi-graph G on n vertices is (k,ℓ)-sparse if every subset of n′⩽n vertices spans at most kn′-ℓ edges. G is tight if, in addition, it has exactly kn-ℓ edges. For integer valuesk and ℓ∈[0,2k), we characterize the (k,ℓ)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)-pebble games. [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs.
Sparse graph, Pebble game, Henneberg sequence, Matroid, Circuit, Rigidity
Lee, Audrey and Streinu, Ileana, "Pebble Game Algorithms and Sparse Graphs" (2008). Computer Science: Faculty Publications, Smith College, Northampton, MA.