On the Minimum Degree up to Local Complementation: Bounds and Complexity

TL;DR

This paper studies the local minimum degree of graphs, providing new bounds for specific graph families and proving the NP-completeness of related decision problems, with implications for quantum computing.

Contribution

It establishes the highest known bound on the local minimum degree for Paley graphs and proves the NP-completeness of the decision problem related to local minimum degree.

Findings

Paley graphs have a local minimum degree greater than sqrt{p} - 3/2.

Existence of infinitely many graphs with linear local minimum degree, up to 0.189 for general graphs.

Deciding the local minimum degree is NP-complete and hard to approximate.

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 NP-complete and that there exists no k-approximation algorithm for this problem for any constant k unless P = NP.