A subset of vertices a graph is minimal if, within all subsets the same size, its vertex boundary minimal. We give complete, geometric characterization sets for planar integer lattice $X$. Our elucidates structure sets, and we are able to use it obtain several applications. show that neighborhood set characterize uniquely $X$: those which congruent any other size. also classify efficient have m...

A secret sharing scheme is a way of protecting a secret by distributing partial information to a set of participants P in such a way that only authorized subsets of P can recover the secret. The family of authorized subsets is called the access structure of the scheme. In 1979, threshold schemes were proposed to realize threshold access structures, and in 1987, multiple assignment schemes were ...

An ideal secret sharing scheme is a method of sharing a secret key in some key space among a finite set of participants in such a way that only the authorized subsets of participants can reconstruct the secret key from their shares which are of the same length as that of the secret key. The set of all authorized subsets of participants is the access structure of the secret sharing scheme. In th...

Minimal inconsistent subsets of knowledge bases play an important role in classical logics, most notably for repair and inconsistency measurement. It turns out that for nonmonotonic reasoning a stronger notion is needed. In this paper we develop such a notion, called strong inconsistency. We show that—in an arbitrary logic, monotonic or not—minimal strongly inconsistent subsets play the same ro...

‎three infinite families of finite abelian groups will be‎ ‎described such that each member of these families has‎ ‎the r'edei $k$-property for many non-trivial values of $k$‎.

Building upon the recent work in communication efficient classical secret sharing schemes of Bitar and El Rouayheb, we study communication efficient quantum secret sharing schemes. In the standard model of quantum secret sharing, typically one is interested in minimal authorized sets for the reconstruction of the secret. In such a setting, reconstruction requires the communication of all the sh...