On Weak Odd Domination and Graph-based Quantum Secret Sharing

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 κ(G), the maximum size of WOD sets of a graph G, and on κ'(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 κ_Q(G) = max(κ(G), n-κ'(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. κ_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 κ_Q i.e. κ_Q(G< 0.811n where n is the order of G). We also prove that deciding for a given graph G whether κ_Q(G)< k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a κ_Q smaller than 0.811n.