Sparse graphs and an augmentation problem

For two integers k>0 and ℓ , a graph G=(V,E) is called (k,ℓ ) -tight if |E|=k|V|-ℓ and i_G(X)≤ k|X|-ℓ for each X⊆ V for which i_G(X)≥ 1 , where i_G(X) denotes the number of induced edges by X . G is called (k,ℓ ) -redundant if G-e has a spanning (k,ℓ ) -tight subgraph for all e∈ E . We consider the following augmentation problem. Given a graph G=(V,E) that has a (k,ℓ ) -tight spanning subgraph, find a graph H=(V,F) with the minimum number of edges, such that G∪ H is (k,ℓ ) -redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is (k,ℓ ) -tight. For general inputs, we give a polynomial algorithm when k≥ℓ and show the NP-hardness of the problem when k<ℓ . Since (k,ℓ ) -tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.