We consider the problem of the existence of uniquely partitionable planar graphs. We survey some recent results and we prove the nonexistence of uniquely (D1,D1)-partitionable planar graphs with respect to the property D1 ”to be a forest”.

Neighborhood is an important concept in covering based rough sets. That under what condition neighborhoods form a partition is a meaningful issue induced by this concept. Many scholars have paid attention to this issue and presented some necessary and sufficient conditions. However, there exists one common trait among these conditions, that is they are established on the basis of all neighborho...

In this paper we tackle the problem of tetrahedralization by breaking non-convex polyhedra into convex subpolyhedra, tetrahedralizing these convex subpolyhedra and merging them together. We generate a Binary Space Partition (BSP) tree from the triangular faces of a polyhedron and use this to identify the convex subpolyhedra in the polyhedron. Each convex subpolyhedron is tetrahedralized individ...

In two previous papers, the study of partitions with short sequences has been developed both for its intrinsic interest and for a variety of applications. The object of this paper is to extend that study in various ways. First, the relationship of partitions with no consecutive integers to a theorem of MacMahon and mock theta functions is explored independently. Secondly, we derive in a succinc...

This paper studies an extension of bi-semiorders in which a “frontier” is added between the various relations used. This extension is motivated by the study of additive representations of ordered partitions and coverings defined on product sets of two components.

Partitioning is a key issue within the Codesign process. As it usually consumes much time, it is important to make it as efficient as possible to avoid costly feedbacks. In this way, this paper describes a partitioning process based on a grouping technique intended to avoid high time overheads due to communications, as well as being able to work with a reliable estimation model.

T paper investigates the effects of a limited consumer memory on the price competition between firms. It studies a specific aspect of memory—namely, the categorization of available price information that the consumers may need to recall for decision making. This paper analyzes competition between firms in a market with uninformed consumers who do not compare prices, informed consumers who compa...

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.

We analyze the moment of inertia I(S), relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if I(S) < I(T). We show that roundest sets of a given size are strongly convex in the discrete sense. Moreover, we introduce the notion of quasi-discs and show that roundest sets are quasi-discs. We use wea...

In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some su...