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, [σ,ρ][σ,ρ]-domination, and perfect codes. We introduce bounds on κ(G)κ(G), the maximum size of WOD sets of a graph G  , and on κ′(G)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 whose threshold is κQ(G)=max⁡(κ(G),n−κ′(G))κQ(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. κQ(G)=n−o(n)κQ(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 κQκQ (i.e. κQ(G)≤0.811nκQ(G)≤0.811n where n is the order of G). We also prove that deciding for a given graph G   whether κQ(G)≤kκQ(G)≤k is NP-complete, which means that one cannot efficiently double check that a graph randomly generated has actually a κQκQ smaller than 0.811n.