Veronika Furst

Nordhaus–Gaddum problems for power domination

Abstract A power dominating set of a graph G is a set S of vertices that can observe the entire graph under the rules that (1) the closed neighborhood of every vertex in S is observed, and (2) if a vertex and all but one of its neighbors are observed, then the remaining neighbor is observed; the second rule is applied iteratively. The power domination number of G , denoted by γ P ( G ) , is the minimum number of vertices in a power dominating set. A Nordhaus–Gaddum problem for power domination is to determine a tight lower or upper bound on γ P ( G ) + γ P ( G ¯ ) or γ P ( G ) ⋅ γ P ( G ¯ ) , where G ¯ denotes the complement of G . The upper and lower Nordhaus–Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement are connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for the domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.

Multiresolution Equivalence and Path Connectedness

An equivalence relation between multiresolution analyses was first introduced in 1996; an analogous definition for generalized multiresolution analyses was given in 2010. This article describes the relationship between the two notions and shows that both types of equivalence classes are path connected in an operator-theoretic sense. The GMRA paths are restricted to canonical GMRAs, and it is shown that whenever two MRAs in L 2(ℝ) are equivalent, the GMRA path construction between their corresponding canonical GMRAs yields the natural analog of the MRA path. Examples are provided of GMRA paths that are distinct from MRA paths.

Characteristic Wavelet Equations and Generalizations of the Spectral Function

Two equations characterize orthonormal (and Parseval) wavelets in L2(ℝ). We trace the history and development of this characterization and present a different argument that uses a generalization of the spectral function of Bownik and Rzeszotnik. By further generalizing this function in the setting of an abstract Hilbert space, we present a compressed proof of the abstract characteristic equation.