A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. In terms of a chess board problem, let Xn be the graph for chess pieceX on the square of side n. Thus, (Xn) is the domination number for che...

Recently, the authors studied the connection between each maximal monotone operator T and a family H(T ) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a uniq...

We are concerned with the poset P(n)=P([1, 2, ..., n]). This is the power set of [n]=[1, 2, ..., n], ordered by inclusion. A set system is simply a subset of P(n). A set system is an antichain if no two of its members are comparable. Conversely a chain is a totally ordered set system. We shall often consider maximal chains; those chains which cannot be extended. In particular such chains contai...

A point in the plane is called a guard for the convex set C if it lies in the interior of C. Let P be a planar point-set. A set S of points is a k-convex guard set for P if every convex k-gon formed from points of P contains a guard from S. We study, for any integer k 3, the minimum size of a k-convex guard set of a given planar point-set of size n. We give the tight bounds for the case k = 3, ...

We consider the task of finding frequent patterns in parallel point processes—also known as finding frequent parallel episodes in event sequences. This task can be seen as a generalization of frequent item set mining: the co-occurrence of items (or events) in transactions is replaced by their (imprecise) co-occurrence on a continuous (time) scale, meaning that they occur in a limited (time) spa...

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in Graphs Combin. 21 (2005), 63–69. A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We prove ...

The aim of this paper is to prove that if a planar set A has a difference set ∆(A) satisfying ∆(A) ⊂ Z + s for suitable s than A has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth...

The paper continues the study of independent set dominating sets in graphs which was started by E. Sampathkumar. A subset D of the vertex set V (G) of a graph G is called a set dominating set (shortly sd−set) in G, if for each set X ⊆ V (G) −D there exists a set Y ⊆ D such that the subgraph 〈X ∪ Y 〉 of G induced by X ∪ Y is connected. The minimum number of vertices of an sd−set in G is called t...

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number k(G), the connected k-domination number c k (G); the k-independent domination number i k (G) and the k-irredundance number irk(G). The authors prove that if an irk-set X is a k-independent set of G, then irk(G) = k(G) = k(G), and that for k ...

We illustrate a process that constructs martingales with help from matrix products that arise naturally in the theory of sampling without replacement. The usefulness of the new martingales is illustrated by the development of maximal inequalities for permuted sequences of real numbers. Some of these inequalities are new and some are variations of classical inequalities like those introduced by ...