In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...

A graph G is said to be distance-balanced if for any edge uv of G, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Let GP (n, k) be a generalized Petersen graph. Jerebic, Klavžar, and Rall [Distance-balanced graphs, Ann. Comb. 12 (2008) 71–79] conjectured that: For any integer k > 2, there exists a positive integer n0 such that the GP (n, k...

A star-factor of a graph G is a spanning subgraph of G such that each component is a star. An edge-weighting of G is a function w : E(G) −→ N, where N is the set of positive integers. Let Ω be the family of all graphs G such that every star-factor of G has the same weight under some fixed edge-weighting w. The open problem of characterizing the class Ω, posed by Hartnell and Rall, is motivated ...

What is the optimal shape of a dendrite? Of course, optimality refers to some particular criterion. In this paper, we look at the case of a dendrite sealed at one end and connected at the other end to a soma. The electrical potential in the fiber follows the classical cable equations as established by W. Rall. We are interested in the shape of the dendrite which minimizes either the attenuation...

Given a connected bipartite graph G, we describe a procedure which enumerates and computes all graphs H (if any) for which there is a direct product factorization G ∼= H × K2. We apply this technique to the problems of factoring even cycles and hypercubes over the direct product. In the case of hypercubes, our work expands some known results by Brešar, Imrich, Klavžar, Rall, and Zmazek [Finite ...

An approach to organ cryopreservation, namely, vitrification has been review ed by FAHY et al.1). RALL and FAHY2) obtained some promissing results using 8-cell mouse embryos indicating the feasibility of obtaining high survival following rapid freezing by vitrification. However, there have been only a preliminary report2) on the feasibility of obtaining offspring by the vitrification method. Th...

Cable theory was introduced for neurons by Wilfrid Rall more than half a century ago, and is widely used today for modeling the voltage and current flow in neuronal and dendritic structures. This theory was derived assuming that the extracellular medium is either inexistent, or modeled as a resistor. For modeling neurons in more realistic situations, where the extracellular medium has more comp...

Let γ {k} t (G) denote the total {k}-domination number of graph G, and let G H denote the Cartesian product of graphs G and H . In this paper, we show that for any graphs G and H without isolated vertices, γ {k} t (G)γ {k} t (H) ≤ k(k + 1)γ {k} t (G H). As a corollary of this result, we have γt (G)γt (H) ≤ 2γt (G H) for all graphs G and H without isolated vertices, which is given by Pak Tung Ho...

protectant for spermatozoa41) led to research on cryopreservation of mammalian embryos. In the past quarter of a century since the first successful cryopreservation of mouse embryos58,60), basic and applied research has resulted in the cryopreservation of embryos of many mammalian species. Improvements have been made in nearly all steps of the cryopreservation process, resulting in considerable...

In this article, we study locating-dominating codes in binary Hamming spaces $$\mathbb {F}^n$$ . Locating-dominating have been widely studied since their introduction 1980s by Slater and Rall. They are dominating sets suitable for distinguishing vertices graphs. Dominating as well multiple articles. Previously, Honkala et al. (Discret Math Theor Comput Sci 6(2):265, 2004) presented a lower boun...