Identifying Codes and Domination in the Product of Graphs

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph $G$ is denoted \gid(G). We consider identifying codes of the direct product $K_n \times K_m$. In particular, we answer a question of Klav\v{z}ar and show the exact value of \gid(K_n \times K_m). It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques \gid(K_n \Box K_n) = \lfloor{\frac{3n}{2}\rfloor}. Letting $m \ge n \ge 2$ be any integers, we show that \IDCODE(K_n \cartprod K_m) = \max\{2m-n, m + \lfloor n/2 \rfloor\}. Furthermore, we improve upon the bounds for \IDCODE(G \cartprod K_m) and explore the specific case when $G$ is the Cartesian product of multiple cliques. Given two disjoint copies of a graph $G$, denoted $G^1$ and $G^2$, and a permutation $\pi$ of $V(G)$, the permutation graph $\pi G$ is constructed by joining $u \in V(G^1)$ to $\pi(u) \in V(G^2)$ for all $u \in V(G^1)$. The graph $G$ is said to be a universal fixer if the domination number of $\pi G$ is equal to the domination number of $G$ for all $\pi$ of $V(G)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture

Domination in Functigraphs

Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $\gamma(G)$ denote the domination number of $G$. It is readily seen that $\gamma(G) \le \gamma(C(G,f)) \le 2 \gamma(G)$. We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.Comment: 18 pages, 8 figure

On regular graphs, V

AbstractLet Γ3 be an infinite regular tree of valence 3. There exist subgroups B of Aut (Γ3) which are 5-regular on Γ3, i.e., sharply transitive on the set of 5-arcs of Γ3. We prove that any two such subgroups are conjugate in Aut (Γ3). The pair (Γ3, B) is a universal 5-regular action in the sense that if (G, A) is a pair consisting of a cubical graph G and a 5-regular subgroup A of automorphisms of G then (G, A) can be “covered” by (Γ3, B) in a certain natural way

A study of the nitrogen content of dune soils, with particular reference to the effects produced by the sea-buckthorn, Hippophae rhamnoides L.

The history of research into the chemistry of dune soils is reviewed, with particular reference to the presence of nitrogen compounds, and also the effects created by the nitrogen-fixing shrub, sea-buckthorn (Hippophaë rhamnoides, L.). Problems associated with the colonization of sand dunes by Hippophaë were investigated in the main study, involving soil sampling at selected sites between December 1980 and November 1981 on the Norfolk coast near Hunstanton. Hippophaë scrub of different ages was chosen, and control sites were left undisturbed. At experimental sites thick polythene sheeting was buried below the surface sand to prevent the downward percolation of solubles. Subsidiary investigations were made of other coastal sites in Britain, some with, and some without, Hippophaë colonies. The soil samples were analysed principally for nitrate ion, but also for ammonium, phosphate, and carbonate ions. The investigations revealed that nitrate levels in dune soil are proportional to the age of the Hippophaë scrub. A seasonal variation in nitrate levels was observed, with peaks in April and September. A model nitrogen economy was proposed to account for this. Statistical tests revealed no significant differences between control and experimental sites suggesting that nitrate is derived principally from Hippophaë rather than the leaf litter or animal products, probably by routine disintegration of the root nodules. The subsidiary sites lacking Hippophaë colonies had no detectable nitrate, or very low levels (< 0.5 ppm) even when heather scrub (Studland Heath, Dorset) or hawthorn scrub (Daymer Bay, Cornwall) was present. This confirms the view that the nitrate detected at Hunstanton came from the Hippophaë scrub. Where Hippophaë was prolific (Gibraltar Point, Lincolnshire) the levels of nitrate detected were higher than at Hunstanton, even in the uncolonized sand of the beach, due possibly to the circulation of nitrates within the ground water of the dune system

A Reckless Verisimilitude: The Archive in James Ellroy’s Fiction

The archive as both plot element and narrative presentation factors significantly into the work of James Ellroy’s novels in the L.A. Quartet and USA Underworld Trilogy series. This article examines the important role of the archive as a source of information and evidence that Ellroy’s characters utilize in their attempts at either maintaining or attacking the status quo. Through these novels, Ellroy conveys the potential power archives wield over the trajectory of history and our understanding of it by demonstrating how the historical record is often shaped in favor of the powerful. Yet even if the archive is a manifestation of the power narratives that dominate society in any given time, it also holds the potential to reveal truths that disrupt that power

Cohesion Metrics for Improving Software Quality

Abstract Software product metrics aim at measuring the quality of software. Modu- larity is an essential factor in software quality. In this work, metrics related to modularity and especially cohesion of the modules, are considered. The existing metrics are evaluated, and several new alternatives are proposed. The idea of cohesion of modules is that a module or a class should consist of related parts. The closely related principle of coupling says that the relationships between modules should be minimized. First, internal cohesion metrics are considered. The relations that are internal to classes are shown to be useless for quality measurement. Second, we consider external relationships for cohesion. A detailed analysis using design patterns and refactorings confirms that external cohesion is a better quality indicator than internal. Third, motivated by the successes (and problems) of external cohesion metrics, another kind of metric is proposed that represents the quality of modularity of software. This metric can be applied to refactorings related to classes, resulting in a refactoring suggestion system. To describe the metrics formally, a notation for programs is developed. Because of the recursive nature of programming languages, the properties of programs are most compactly represented using grammars and formal lan- guages. Also the tools that were used for metrics calculation are described.Siirretty Doriast