On the Minimum Degree Up to Local Complementation: Bounds and Complexity
نویسندگان
چکیده
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 √ 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 l-approximation algorithm for this problem for any constant l unless P = NP .
منابع مشابه
Minimum Degree Up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms
The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3 8 n + o(n) and n 4 + o(n) for bi...
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