On Metric Dimension of Functigraphs

The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. 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 $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=f(u)\}$. We study how metric dimension behaves in passing from $G$ to $C(G,f)$ by first showing that $2 \le \dim(C(G, f)) \le 2n-3$, if $G$ is a connected graph of order $n \ge 3$ and $f$ is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.Comment: 10 pages, 7 figure

Cheminement des carrières de direction dans la fonction publique au Canada

Cet article présente un résumé d'une étude entreprise auprès des hauts fonctionnaires dans la fonction publique du Canada. Elle porte sur le cheminement des carrières et touche aussi à la mobilité, à l'occupation et aux sources de recrutement.Who are the most capable executives in the Federal Public Service ? What are their experience characteristics ? How mobile have they been ? What are their academic backgrounds ? How does age relate to seniority and earning power ?These and related questions must be answered if manpower at upper government levels is to be managed more effectively.A fourteen-month study recently completed by the Personnel Consulting Division, Bureau of Management Consulting Services, Ottawa, has made a major contribution towards providing these answers.In the Public Service of Canada, at the time the study was made in July, 1965, there were some 625 executives with salaries ranging from $16,000 to$30,000. These are the men and women who fill the top three or four levels in each department or agency. Officially, they are known as Senior Officers 1, 2, 3. and Deputy Ministers (Deputy Heads).Data on 591 of these executives was obtained from files and various other sources, coded and fed into a Bendix G-20 computer along with some 40 questions, the answers to which provided the following information :The typical executive is 51 years old and probably has a B.A. in Social Management Sciences (Arts, History, Political Science, Commerce, Sociology, etc.). While he could have graduated from any university in any province, the chances are greatest that he obtained his degree from the University of Toronto. He has had military service, reads a second language and joined the Public Service at a junior managerial level at 34 years of age, after having worked outside the Public Service for 11 years.At the time of his appointment to the executive level, he was 46 years of age and had 23 years of working experience. As of July 1, 1967, he had 28 years working experience and was earning $21,000 a year.BASIC PERSONAL DATAIn considering the personal data in more detail, it was found that ages ranged from 30 to 69, with an average age of 51.2. The average ages for the four levels (Senior Officer 1, 2, 3, and Deputy Minister — Deputy Head), were 50.3 52.6, 51.7 and 54.9 years respectively. The similarity of the averages implies career progression problems and also highlights the serious executive replacement problem the government faces in the next 10-15 years.A relatively high level of education was found among the government's top executives. 81$14,000 plus). Close to 20% entered directly at the executive level.SALARYAs of July 1, 1967, the average executive in the Public Service of Canada was earning $20,927 a year.It should be noted that this is a total figure because Public Servants do not receive bonusses or stock options, and there is not profit to be « profit-shared ».For the three career path categories they were :Public Service Only$21,297Private-Public 20,861Public-Private-Public 20,862Subsequent to this study, the government released proposed salary scales for the executive levels ranging from a maximum of $23,500 for Senior Executive Officers 1 up to a maximum of$40,000 for certain Deputy Ministers.APPOINTMENT TO EXECUTIVE LEVELThe average age at time of appointment to the Senior Officer 1 level was 45.6, to the SO 2 level 48, SO 3 level 47 and DM-DH 52. The similarity of these figures is interesting, since one would expect an executive to progress from the lowest to highest level. Also, based on today's standards, the ages would appear to be rather high.TRENDS IN AGE AT APPOINTMENTIt is generally believed that there has been a trend to appoint younger officers to the executive levels, but the study casts some doubt on the belief, certainly as far as government is concerned. However, more recent data obtained since the completion of the study indicates a change in the trend with the average ages of Senior Officers 1 appointed in 1966, 1967 and 1968, decreasing from 48 to 46 to 43 years respectively.EXPERIENCEThe average government executive has 28 years experience, almost two-thirds of which will have been spent in the Public Service. He will have spent about 15% in the category we called « Business and Self-Employed » and 10% in military service.However, there were indications that the more time executives spent in industry the less capable they were of adjusting to the political and Public Service atmosphere. Similarly it was found that extended career military service tended to reduce the rate of progress.RECRUITMENTPrior to July, 1967, about 76% of all appointments to the executive level were made from within the individuals present department and about 9% were from other departments. Therefore, some 85% of all appointments were from within the Public Service. Of the remaining 15%, some 10% came from Industry and 5% were recruited from Provincial Governments (2.4%), Universities (1.5%) and the Armed Forces (.3%).MOBILITYWithin the Public Service, a great interest has recently developed in the movement, or lack of movement, of management level personnel from one department to another. Traditionally, Public Servants have tended to remain in one department, rising to senior levels within what many claim to be a narrow occupational field.From the findings it appeared that moving from one department to another, within the Public Service, tended to improve the possibilities of achieving more rapid progress.A further analysis of movement after reaching the executive level showed that slightly more than 12% of all executives changed departments at least once after reaching this level and 1.3% had changed twice. This figure is changing rapidly as over 30% of the appointments made in 1968, at the executive level, resulted in movement between departments

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

Continuity of heavy Rydberg behaviour in the ungerade ion-pair states of H 2

Heavy Rydberg behaviour and absolute quantum defects are reported for resonances in the ungerade manifold of H2 above the (1s, 3l) dissociation limit. The continuity of the vibrational progression of the B\u27\u27B-bar state through the crossing with the 3p asymptote is demonstrated and a predominantly diabatic picture of the vibrational motion emerges, indicating that the ion-pair resonances possess little 61Σu+ state character

Pilot In-Trail Procedure Validation Simulation Study

A Human-In-The-Loop experiment was conducted at the National Aeronautics and Space Administration (NASA) Langley Research Center (LaRC) to investigate the viability of the In-Trail Procedure (ITP) concept from a flight crew perspective, by placing participating airline pilots in a simulated oceanic flight environment. The test subject pilots used new onboard avionics equipment that provided improved information about nearby traffic and enabled them, when specific criteria were met, to request an ITP flight level change referencing one or two nearby aircraft that might otherwise block the flight level change. The subject pilots subjective assessments of ITP validity and acceptability were measured via questionnaires and discussions, and their objective performance in appropriately selecting, requesting, and performing ITP flight level changes was evaluated for each simulated flight scenario. Objective performance and subjective workload assessment data from the experiment s test conditions were analyzed for statistical and operational significance and are reported in the paper. Based on these results, suggestions are made to further improve the ITP

In-Trail Procedure Air Traffic Control Procedures Validation Simulation Study

In August 2007, Airservices Australia (Airservices) and the United States National Aeronautics and Space Administration (NASA) conducted a validation experiment of the air traffic control (ATC) procedures associated with the Automatic Dependant Surveillance-Broadcast (ADS-B) In-Trail Procedure (ITP). ITP is an Airborne Traffic Situation Awareness (ATSA) application designed for near-term use in procedural airspace in which ADS-B data are used to facilitate climb and descent maneuvers. NASA and Airservices conducted the experiment in Airservices simulator in Melbourne, Australia. Twelve current operational air traffic controllers participated in the experiment, which identified aspects of the ITP that could be improved (mainly in the communication and controller approval process). Results showed that controllers viewed the ITP as valid and acceptable. This paper describes the experiment design and results

Bend it like Beckham: embodying the motor skills of famous athletes.

Observing an action activates the same representations as does the actual performance of the action. Here we show for the first time that the action system can also be activated in the complete absence of action perception. When the participants had to identify the faces of famous athletes, the responses were influenced by their similarity to the motor skills of the athletes. Thus, the motor skills of the viewed athletes were retrieved automatically during person identification and had a direct influence on the action system of the observer. However, our results also indicated that motor behaviours that are implicit characteristics of other people are represented differently from when actions are directly observed. That is, unlike the facilitatory effects reported when actions were seen, the embodiment of the motor behaviour that is not concurrently perceived gave rise to contrast effects where responses similar to the behaviour of the athletes were inhibited

Vertex-Coloring with Star-Defects

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors