On weak odd domination and graph-based quantum secret sharing

Abstract

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 ${\kappa }^{\prime }(G)$, 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 ${\kappa }_Q(G)=\max (\kappa (G),n-{\kappa }^{\prime }(G))$. 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. ${\kappa }_Q(G)\le 0.811n$ where n is the order of G). We also prove that deciding for a given graph G whether ${\kappa }_Q(G)\le k$ 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.