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 ...

We show that, for any n-vertex graph G and integer parameter k, there are at most 34k−n4n−3k maximal independent sets I ⊂ G with |I| ≤ k, and that all such sets can be listed in time O(34k−n4n−3k). These bounds are tight when n/4 ≤ k ≤ n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 3/4)) ≈ 2.4150, improving a previous O((1 + 3)) ≈ 2.4422 alg...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γtr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then γtr(T ) ≥ d(n + 2)/2e. Moreover, we s...

In this paper, we consider graphs having a unique minimum independent dominating set. We first discuss the effects of deleting a vertex, or the closed neighborhood of a vertex, from such graphs. We then discuss five operations which, in certain circumstances, can be used to combine two graphs, each having a unique minimum independent dominating set, to produce a new graph also having a unique m...

Teichmüller’s classical mapping problem for plane domains concerns finding a lower bound for the maximal dilatation of a quasiconformal homeomorphism which holds the boundary pointwise fixed, maps the domain onto itself, and maps a given point of the domain to another given point of the domain. For a domain D ⊂ Rn , n ≥ 2 , we consider the class of all Kquasiconformal maps of D onto itself with...