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

对于两个整数\(k>0\)和\(\ell \)，一个图\(G=(V,E)\)被称为\((k,\ell )\) -tight if \(|E |=k|V|-\ell \)和\(i_G(X)\le k|X|-\ell \)对于每个\(X\subseteq V\)其中\(i_G(X)\ge 1 \)，其中\(i_G(X)\)表示X的诱导边数。如果\(Ge\)具有所有\(e\in E\)的跨度\((k,\ell )\) -紧子图，则G被称为\((k,\ell )\) -冗余。我们考虑以下增强问题。给定一个图\(G=(V,E)\)具有\（（K，\ ELL）\） -tight生成子图，找到的曲线图\（H =（V，F）\）与边缘的最小数量，使得\（G \座H \）是\((k,\ell )\) - 冗余。当输入为\((k,\ell )\) -tight时，我们为这个增强问题给出了一个多项式算法和一个最小-最大定理。对于一般输入，我们在\(k\ge \ell \)时给出多项式算法，并在\(k<\ell \)时显示问题的 NP-hardness 。由于\((k,\ell )\) -tight 图在刚性理论中起着重要作用，因此这些算法可用于通过添加最小的一组新条来使几种类型的刚性框架具有冗余刚性。