A characterization of robert's inequality for boxicity

AbstractF.S. Roberts defined the boxicity of a graph G as the smallest positive integer n for which there exists a function F assigning to each vertex x ϵG a sequence F(x)(1),F(x)(2),…, F(x)(n) of closed intervals of R so that distinct vertices x and y are adjacent in G if and only if F(x)(i)∩F(y)(i)≠∅ for i = 1, 2, 3, …, n. Roberts then proved that if G is a graph having 2n + 1 vertices, then the boxicity of G is at most n. In this paper, we provide an explicit characterization of this inequality by determining for each n ⩾ 1 the minimum collection Cn of graphs so that a graph G having 2n + 1 vertices has boxicity n if and only if it contains a graph from Cn as an induced subgraph. We also discuss combinatorial connections with analogous characterization problems for rectangle graphs, circular arc graphs, and partially ordered sets

Support theorems in abstract settings

In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain space, we allow algebraic structures equipped with families of algebraic operations whose operations are mutually distributive with respect to each other. We introduce several new concepts in such algebraic structures, the notions of convex set, extreme set, and interior point with respect to a given family of operations, furthermore, we describe their most basic and required properties. In the context of the range space, we introduce the notion of completeness of a partially ordered set with respect to the existence of the infimum of lower bounded chains, we also offer several sufficient condition which imply this property. For instance, the order generated by a sharp cone in a vector space turns out to possess this completeness property. By taking several particular cases, we deduce support and extension theorems in various classical and important settings

About Nonstandard Neutrosophic Logic (Answers to Imamura 'Note on the Definition of Neutrosophic Logic')

In order to more accurately situate and fit the neutrosophic logic into the framework of nonstandard analysis, we present the neutrosophic inequalities, neutrosophic equality, neutrosophic infimum and supremum, neutrosophic standard intervals, including the cases when the neutrosophic logic standard and nonstandard components T, I, F get values outside of the classical real unit interval [0, 1], and a brief evolution of neutrosophic operators. The paper intends to answer Imamura criticism that we found benefic in better understanding the nonstandard neutrosophic logic, although the nonstandard neutrosophic logic was never used in practical applications.Comment: 16 page