For two integers $k>0$ and $\ell$, a graph $G=(V,E)$ is called $(k,\ell)$-tight if $|E|=k|V|-\ell$ and $i_G(X)\leq k|X|-\ell$ for each $X\sbse V$ that induces at least one edge.
$G$ is called $(k,\ell)$-sparse 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 $(k,\ell)$-sparse.
In this paper, 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 $k\geq\ell$ 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.
Bibtex entry:
@techreport{egres-20-06,
AUTHOR | = | {Kir{\'a}ly, Csaba and Mih{\'a}lyk{\'o}, Andr{\'a}s}, |
TITLE | = | {Sparse graphs and an augmentation problem}, |
NOTE | = | {{\tt egres.elte.hu}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2020}, |
NUMBER | = | {TR-2020-06} |