L. Guibas, N. Milosavljevic and A. Motskin. Connected Dominating Sets on Dynamic Geometric Graphs, Canadian Conference on Computational Geometry (CCCG), 2010.

Abstract:

We propose algorithms for efficiently maintaining a constant-approximate minimum connected dominating set (MCDS) of a geometric graph under node insertions and deletions. Assuming that two nodes are adjacent in the graph iff they are within a fixed geometric distance, we show that an $O(1)$-approximate MCDS of a graph in $R^d$ with $n$ nodes can be maintained with polylogarithmic (in $n$) work per node insertion/deletion as compared with $Omega(n)$ work to maintain the optimal MCDS, even in the weaker kinetic setting. In our approach, we ensure that a topology change caused by inserting or deleting a node only affects the solution in a small neighborhood of that node, and show that a small set of range queries and bichromatic closest pair queries is then sufficient to efficiently repair the CDS.

Bibtex: 

@inproceedings{gmm-cdsdgg-10,
    author={Leonidas J. Guibas and Nikola Milosavljevic and Arik Motskin},
    title={Connected Dominating Sets on Dynamic Geometric Graphs},
    booktitle={Proc. of the 22nd Canadian Conference on Computational Geometry (CCCG)},
    year={2010}
}