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 such that ∃D ⊆ V \B, ∀v ∈ B, |N(v)∩D| = 1 mod 2. We point out the connections of weak odd domination with odd domination, (σ, ρ)-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 protocol introduced by Markham and Sanders [9]. Indeed, a graph G of order n can be used to define a quantum secret sharing protocol where κQ(G) = max(κ(G), n − κ(G)) is a threshold ensuring that any set of more than κQ(G) players can recover a quantum secret. We show the hardness of finding the optimal threshold of a graphbased quantum secret sharing protocol. Finally, using probabilistic methods, we show the existence of an infinite family of graphs {Gi} with ‘small’ κQ, i.e. such that κQ(Gi) ≤ 0.811ni where ni is the order of Gi, and that with high probability a random graph G of order n satisfies κQ(G) ≤ 0.87n.