A graph $G=(V,E)$ is called $(k,\ell)$-tight if $|E(X)|\leq k|X|-\ell$ for all $X\sbse V$ with $|X|\geq 2$ and $|E|=k|V|-\ell$.
A graph $G$ 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 minimum number of edges, such that $G+H$ is $(k,\ell)$-redundant.
In this paper, we give a polynomial algorithm for this augmentation problem for $k\geq \ell$. We also give a polynomial algorithm for the case where $G$ is a $(k,\ell)$-tight graph with $k<\ell\leq \frac{3}{2}k$. As $(k,\ell)$-tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of bar-and-joint frameworks redundantly rigid by adding a smallest set of new bars.
Bibtex entry:
@techreport{egres-15-03,
AUTHOR | = | {Kir{\'a}ly, Csaba}, |
TITLE | = | {Rigid graphs and an augmentation problem}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2015}, |
NUMBER | = | {TR-2015-03} |