Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...

Given an embedding f: G -+ 2 of a graph G in the two-dimensional lattice, let Iff be the maximum L1 distance between points f(x) and f(y) where xy is an edge of G. Let B2 ( G) be the minimum Ifl over all embeddings f. It is shown that the determination of B2 ( G ) for arbitrary G is NP-complete. Essentially the same proof can be used in showing the NP-completeness of minimizing If I over all em...

We de ne the notions of reducibility and completeness in (two party and multi-party) private computations. Let g be an n-argument function. We say that a function f is reducible to a function g if n honest-but-curious players can compute the function f n-privately, given a black-box for g (for which they secretly give inputs and get the result of operating g on these inputs). We say that g is c...

We define the notions of reducibility and completeness in multi-party private computations. Let g be an n-argument function. We say that a function f is reducible to g if n honest-but-curious players can compute the function f nprivately, given a black-box for g (for which they secretly give inputs and get the result of operating g on these inputs). We say that g is complete (for multi-party pr...

This paper introduces a semantic analysis of the Rely-Guarantee (R-G) approach to the compositional verification of shared-variable concurrency. The main contribution is a new completeness proof.

We introduce a family of hypergroups, called weakly complete, generalizing the construction complete hypergroups. Starting from given group G, our prescribes β-classes hypergroups and allows some hyperproducts not to be parts, based on suitably defined relation over G. The commutativity degree can related that underlying group. Furthermore, in analogy commutativity, we completeness finite analy...

For a graph G=(V,E) with V=V(G) and E=E(G), perfect double Italian dominating function is f:V→{0,1,2,3} having the property that 3≤∑u∈NG[v]f(u)≤4, for every vertex v∈G f(v)∈{0,1}. The weight of f sum f(V)=∑v∈V(G)f(v) minimum on G domination number γdIp(G) G. We initiate study functions. check γdIp some standard graphs evaluate γdI such graphs. functions versus Roman are perused. NP-completeness...