We show that the qualitative form of the prime k-tuple conjecture implies a certain conjecture related to the least size of the interval containing exactly ` prime numbers. Also, we prove that the latter conjecture implies the prime k-tuple conjecture for some minimal admissible set. The proof is self-contained and elementary.

Dai, Li, and Wu [11] [30] proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. In this paper we consider the “average case”performance of Rule k for the model of random unit disk graphs constructed from n random points in an `n × `n square. We show that if k ≥ 3 and `n = o(√n), then the expected size of the Rule k dominating set...

In this paper, we study the Dominating Set problem in random graphs. In a random graph, each pair of vertices are joined by an edge with a probability of p, where p is a positive constant less than 1. We show that, given a random graph in n vertices, a minimum dominating set in the graph can be computed in expected 2 2 2 n) time. For the parameterized dominating set problem, we show that it can...

A subset S of the vertices of a graph G is an outer-connected dominating set, if S is a dominating set of G and G − S is connected. The outer-connected domination number of G, denoted by γ̃c(G), is the minimum cardinality of an OCDS of G. In this paper we generalize the outer-connected domination in graphs. Many of the known results and bounds of outer-connected domination number are immediate c...

Phylogenies are useful for organizing knowledge of biological diversity, for structuring classifications, and for providing knowledge of events that occurred during evolution. Different phylogenetic reconstruction techniques are available. In this paper Distance based technique is used. Distance measure is an important issue in phylogenetic analysis. Traditional approaches are time-consuming du...

This paper investigates a tuple-oriented approach to tabulation. In this approach solutions to a call are stored and computed one at a time, which results in tables contain incomplete sets of solutions. Since the answers stored in the tables are partial sets of solutions to a call, a subgoal that is an instance of a stored call can require an answer that is not among the stored solutions. Conse...

Rule k (k=1,2,3,...) is a well known family of approximation algorithms that can be used to find connected dominating sets in a graph. They were originally proposed by Dai, Li, and Wu as the basis for efficient routing methods for ad hoc wireless networks. In this paper we study the asymptotic performance of Rules 1 and 2 on random unit disk graphs formed from n random points in an `n × `n squa...

In this paper we introduce a novel, simpler form of the polytope of inner Bayesian approximations of a belief function, or “consistent probabilities”. We prove that the set of vertices of this polytope is generated by all possible permutations of elements of the domain, mirroring a similar behavior of outer consonant approximations. An intriguing connection with the behavior of maximal outer co...

In wireless sensor networks, virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem. In this paper we consid...