On the Minimum Degree up to Local Complementation: Bounds and Complexity
Abstract
The local minimum degree of a graph is the minimum degree reached by means of a series of local complementations. In this paper, we investigate on this quantity which plays an important role in quantum computation and quantum error correcting codes. First, we show that the local minimum degree of the Paley graph of order p is greater than sqrt{p}  3/2, which is, up to our knowledge, the highest known bound on an explicit family of graphs. Probabilistic methods allows us to derive the existence of an infinite number of graphs whose local minimum degree is linear in their order with constant 0.189 for graphs in general and 0.110 for bipartite graphs. As regards the computational complexity of the decision problem associated with the local minimum degree, we show that it is NPcomplete and that there exists no kapproximation algorithm for this problem for any constant k unless P = NP.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 arXiv:
 arXiv:1204.4564
 Bibcode:
 2012arXiv1204.4564J
 Keywords:

 Computer Science  Computational Complexity;
 Quantum Physics
 EPrint:
 11 pages