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For two integers $$k>0$$ k > 0 and $$\ell $$ ℓ , a graph $$G=(V,E)$$ G = ( V , E ) is called $$(k,\ell )$$ ( k , ℓ ) -tight if $$|E|=k|V|-\ell $$ | E | = k | V | - ℓ and $$i_G(X)\le k|X|-\ell $$ i G ( X ) ≤ k | X | - ℓ for each $$X\subseteq V$$ X ⊆ V for which $$i_G(X)\ge 1$$ i G ( X ) ≥ 1 , where $$i_G(X)$$ i G ( X ) denotes the number of induced edges by X. G is called $$(k,\ell )$$ ( k , ℓ ) -redundant if $$G-e$$ G - e has a spanning $$(k,\ell )$$ ( k , ℓ ) -tight subgraph for all $$e\in E$$ e ∈ E . We consider the following augmentation problem. Given a graph $$G=(V,E)$$ G = ( V , E ) that has a $$(k,\ell )$$ ( k , ℓ ) -tight spanning subgraph, find a graph $$H=(V,F)$$ H = ( V , F ) with the minimum number of edges, such that $$G\cup H$$ G ∪ H is $$(k,\ell )$$ ( k , ℓ ) -redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is $$(k,\ell )$$ ( k , ℓ ) -tight. For general inputs, we give a polynomial algorithm when $$k\ge \ell $$ k ≥ ℓ and show the NP-hardness of the problem when $$k<\ell $$ k < ℓ . Since $$(k,\ell )$$ ( 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.

The two main concepts of Rigidity Theory are rigidity, where the framework has no continuous deformation, and global rigidity, where the given distance set determines the locations of the points up to isometry. We consider the following augmentation problem. Given a minimally rigid graph G = (V, E) in R 2 , find a minimum cardinality edge set F such that the graph G = (V, E + F ) is globally rigid in R 2 . We provide a min-max theorem and an O(|V | 2 ) time algorithm for this problem.

We consider the following augmentation problem: Given a rigid graph G = (V, E), find a minimum cardinality edge set F such that the graph G = (V, E ∪ F ) is globally rigid. We provide a min-max theorem and a polynomial-time algorithm for this problem for several types of rigidity, such as rigidity in the plane or on the cylinder. Rigidity is often characterized by some sparsity properties of the underlying graph and global rigidity is characterized by redundant rigidity (where the graph remains rigid after deleting an arbitrary edge) and 2-or 3-vertex-connectivity. Hence, to solve the above-mentioned problem, we define and solve polynomially a combinatorial optimization problem family based on these sparsity and connectivity properties. This family also includes the problem of augmenting a k-tree-connected graph to a highly k-tree-connected and 2connected graph. Moreover, as an interesting consequence, we give an optimal solution to the so-called global rigidity pinning problem, where we aim to find a minimum cardinality vertex set X for a rigid graph G = (V, E), such that the graph G + K X is globally rigid in R 2 where K X denotes the complete graph on the vertex set X.

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