  A weak odd dominated (WOD) set in a graph is a subset B of vertices for which
there exists a distinct set of vertices C such that every vertex in B has an
odd number of neighbors in C. We point out the connections of weak odd
domination with odd domination, [sigma,rho]-domination, and perfect codes. We
introduce bounds on \kappa(G), the maximum size of WOD sets of a graph G, and
on \kappa'(G), the minimum size of non WOD sets of G. Moreover, we prove that
the corresponding decision problems are NP-complete. The study of weak odd
domination is mainly motivated by the design of graph-based quantum secret
sharing protocols: a graph G of order n corresponds to a secret sharing
protocol which threshold is \kappa_Q(G) = max(\kappa(G), n-\kappa'(G)). These
graph-based protocols are very promising in terms of physical implementation,
however all such graph-based protocols studied in the literature have
quasi-unanimity thresholds (i.e. \kappa_Q(G)=n-o(n) where n is the order of the
graph G underlying the protocol). In this paper, we show using probabilistic
methods, the existence of graphs with smaller \kappa_Q (i.e. \kappa_Q(G)<
0.811n where n is the order of G). We also prove that deciding for a given
graph G whether \kappa_Q(G)< k is NP-complete, which means that one cannot
efficiently double check that a graph randomly generated has actually a
\kappa_Q smaller than 0.811n.

Computational Complexity;Quantum Physics