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'(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.
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https://hal.archives-ouvertes.fr/hal-00944428
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Submitted on : Monday, February 10, 2014 - 3:53:40 PM
Last modification on : Friday, October 25, 2019 - 2:01:19 AM

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  • HAL Id : hal-00944428, version 1
  • ARXIV : 1112.2495

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Sylvain Gravier, Jérôme Javelle, Mehdi Mhalla, Simon Perdrix. On Weak Odd Domination and Graph-based Quantum Secret Sharing. 2012. ⟨hal-00944428⟩

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