  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 3n/8+o(n) and n/4+o(n)
for bipartite graphs, improving the known n/2 upper bound. We also prove that
the local minimum degree is smaller than half of the vertex cover number (up to
a logarithmic term). The local minimum degree problem is NP-Complete and hard
to approximate. We show that this problem, even when restricted to bipartite
graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which
W[1]-hardness is a long standing open question. Finally, we show that the local
minimum degree is computed by a O*(1.938^n)-algorithm, and a
O*(1.466^n)-algorithm for the bipartite graphs.
